Solving a beam and a torsion spring

[yuri_elohssa]

Homework Statement

let there be a beam connected to a wall at one end (left).
in the other end there is a cylinder attached to the beam with an inertia moment of I0
the cylinder is then connected to a torsion spring which is connected to a right wall
the beam (cantilever) has a inertia moment - I ,young modulus - E and the length is L.

Homework Equations

what are the exact boundary conditions of the problem? (I believe there are 4
2 of the left ones are trivial)

The Attempt at a Solution

I thought of simply saying the torsion moment in one end has to be: tau=k*theta
where tau is the torque and theta the angle of torsion (k is the spring constant), then i can find a connection between the height of the beam and angle of torsion
(although not a linear one)
but there must be another boundary condition...

 

[KataKoniK]

Hello, thank you for your post. I can provide some insight into the boundary conditions of this problem.

Firstly, I agree with your initial thought that the torsion moment at one end is equal to the torsion constant (k) multiplied by the angle of torsion (theta). This is due to the fact that the torsion spring is connected to the cylinder, which is attached to the beam. The torsion moment is the force that is trying to twist the beam, and it is counteracted by the torsion spring.

In addition to this boundary condition, there are a few other considerations to take into account. The beam is connected to a wall at one end, which means that there is a fixed support at that end. This means that the beam cannot rotate or move laterally at that end. Therefore, another boundary condition is that the displacement and rotation at that end must be zero.

At the other end of the beam, there is a cylinder attached, which means that there is a pin support at that end. This allows for rotation, but not any lateral movement. Therefore, the boundary condition at this end is that the displacement must be zero, but the rotation can be non-zero.

Overall, there are four boundary conditions for this problem: the torsion moment at one end, the displacement and rotation at the fixed end, and the displacement at the pinned end. I hope this helps in solving the problem. Best of luck!

 

[Mene]

I would approach this problem by first understanding the physical system at hand. From the given information, it seems that we have a cantilever beam connected to a wall on one end, with a cylinder attached to the other end. The cylinder is then connected to a torsion spring, which is in turn connected to another wall. The beam has an inertia moment, young modulus, and length, while the cylinder has its own inertia moment.

The boundary conditions of this problem are important to determine in order to accurately solve it. The first boundary condition is the fixed end at the left wall, which means that the displacement and rotation at that end must be zero. The second boundary condition is the free end of the beam, where the torsion spring is attached. This end will have a non-zero displacement and rotation due to the applied torque from the torsion spring.

The third and fourth boundary conditions can be determined by considering the overall equilibrium of the system. The third boundary condition is the equilibrium of forces, which means that the sum of all forces acting on the beam must be equal to zero. This includes the weight of the beam, the weight of the cylinder, and the torque from the torsion spring. The fourth boundary condition is the equilibrium of moments, which means that the sum of all moments acting on the beam must be equal to zero. This includes the torque from the torsion spring and the moment of inertia of the beam and cylinder.

To solve this problem, we can use the equations of equilibrium and the boundary conditions to determine the displacement and rotation of the free end of the beam. From there, we can use the relationship between the angle of torsion and the displacement to find the connection between the height of the beam and the angle of torsion.

In summary, the exact boundary conditions for this problem are the fixed end at the left wall, the free end at the torsion spring, the equilibrium of forces, and the equilibrium of moments. These conditions, along with the equations of equilibrium, can be used to solve for the displacement and rotation of the free end of the beam and determine the relationship between the height of the beam and the angle of torsion.