Power transmission of a multi-stepped shaft

[jasonnaylor]

Homework Statement

Imagine a stepped turbine shaft for a wind turbine.

(See PDF 'Question' for image)
Wind hits the turbine blades with a pressure P Pa in a direction which may be averaged out at 90 degrees to the y-axis, creating a torque Ta, which acts clockwise in the x-direction as we look at the diagram from the left hand side.

We may assume that the transmission shaft L1, L2 and L3 is the same material machined to different diameters d1, d2 and d3 respectively, having a young’s modulus of E (N/m2), and a rigidity modulus of G(N/m2).

The armature housing contains the armature and associated electrical system to generate current for the grid. This armature is housed on bearings having a damping constant of c (N1/2), which provides a resistive torque to the system.
[/B]
a. Design an equation that describes the power transmission over the length of the shaft up to and including the armature housing. Assume that the blades of the turbine may be modeled as a disc with Mass M , radius r and gyration radius k. Ignore the fillet radii between shafts and simply treat as a 90 degree turn.b. Derive an equation that describes the maximum displacement of the centre of rotation of the shaft turbine blades from the axis “XX” assuming that the transmission shaft has a weight of W Newtons. State clearly in your analysis any assumptions that you have made.c. A block of ice having a mass β falls from a plane and hits the blades at an angle of https://www.physicsforums.com/file:///C:/Users/jason/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png to the vertical y-axis. What happens to the transmission shaft? Explain analytically in detail stating any assumptions that you have made.

Homework Equations

P= T.ω[/B]

The Attempt at a Solution

(See attached PDF 'solution') I've been trying to crack this question for days and days and I am going knowhere.[/B]

 

[KataKoniK]

Thank you for your question. I would like to offer some help in solving this problem.

Firstly, for part (a), we can use the equation P=Tω, where P is the power transmitted, T is the torque, and ω is the angular velocity. We can assume that the wind turbine blades are a disc with mass M, radius r, and gyration radius k. Therefore, the moment of inertia of the blades can be calculated as I=1/2Mr^2+1/2Mk^2. The torque acting on the blades can be calculated as Ta=P/(ωsinθ), where θ is the angle between the wind direction and the y-axis. This torque is transmitted through the shaft to the armature housing.

To calculate the power transmitted through the shaft, we can use the equation P=Tω. We can assume that the shaft is a continuous beam with a varying cross-sectional area. Therefore, we can use the equation for power transmitted through a beam, P=Tω=2π∫M(x)ω(x)dx, where M(x) is the bending moment and ω(x) is the angular velocity at a distance x from the origin. We can determine the bending moment using the equation M(x)=T(x)/L, where T(x) is the torque at a distance x from the origin and L is the length of the shaft. We can then substitute this into the equation for power and integrate over the length of the shaft to obtain the total power transmitted.

For part (b), we can use the equation M=Iα, where M is the moment acting on the shaft, I is the moment of inertia, and α is the angular acceleration. We can assume that the shaft has a weight of W Newtons and is rotating at an angular velocity of ω. Therefore, the moment acting on the shaft can be calculated as M=Wxω, where x is the displacement of the centre of rotation from the axis "XX". We can then substitute this into the equation M=Iα and solve for x to obtain the maximum displacement of the centre of rotation.

For part (c), we can assume that the block of ice has a mass β and is falling at an angle of θ to the vertical y-axis. When it hits the blades, it will exert a force on the blades in the direction of the wind. This force will cause the blades