Vibrating uprigt cantilever beam

[AsgerJon]

Suppose a cantilever beam stands uprigt, such as a tube attached to the floor in one end and free at the other end. Suppose then that a force is applied normal to the beam at some point along the beam causing a deflection. Then suppose that at t=0 the force disappears, and the intertia of the beam as its deflection diminishes causes an underdamped motion.

The question I cannot figure out is how to accurately determine a second order differential equation, which will describe deflection as a function of time.

All I have is the idea of assuming no damping, but then I end up with harmonic vibrations, which never change. That which I'm missing from my equation is a way to determine the damping constant or function in my second order differential equation, such that the damping force is proportional to the velocity of the beam at a given time, and is in opposite direction to the motion of the beam.

The beam is assumed to have uniform E and I.

How do I determine the damping effect theoretically?

 

[eljose79]

Hello,

Thank you for your question. I can help you understand how to determine the damping effect theoretically for this scenario.

Firstly, let's define some terms for better understanding. Damping is a force that opposes the motion of a vibrating object, in this case, the cantilever beam. It is caused by internal friction and external resistances. The damping force is proportional to the velocity of the beam and acts in the opposite direction to the motion.

To accurately determine a second-order differential equation for this scenario, we need to consider the forces acting on the beam. These include the applied force, the damping force, and any restoring forces (such as the elastic force due to the deflection of the beam).

The equation of motion for the cantilever beam can be written as:

m*x'' + c*x' + k*x = F(t)

Where m is the mass of the beam, c is the damping constant, k is the stiffness constant, x is the deflection of the beam, and F(t) is the applied force.

To determine the damping constant, we need to consider the energy dissipation in the system. As the beam vibrates, energy is continuously lost due to the damping force. This energy loss is proportional to the velocity of the beam and the damping constant. Therefore, we can write the damping force as:

F_d = -c*x'

Where c is the damping constant and x' is the velocity of the beam.

To determine the value of the damping constant, we can use experimental data or theoretical calculations. For experimental data, we can measure the damping ratio (ζ) of the system, which is the ratio of the damping force to the elastic force. This can be done by measuring the amplitude of the beam's motion over time and then using the formula ζ = c/2*sqrt(km). Once we have the damping ratio, we can calculate the damping constant using the formula c = 2*sqrt(km)*ζ.

For theoretical calculations, we can use the material properties of the beam (such as the Young's modulus and the moment of inertia) and the geometry of the beam to determine the damping constant. This can be done by considering the energy dissipation in the system due to internal friction and external resistances.

In conclusion, to accurately determine the damping effect theoretically, we need to consider the forces acting on the beam and the energy dissipation in the system. The damping