PHYSM-The new wind turbine will adopt a standard t

The new wind turbine will adopt a standard three-bladed configuration and for economic reasons it was decided to use a pre-existing blade design with a length of 10 m. This section will focus on the structural design of the main axle or shaft as shown in Figure 0. a) Assume that, including safety factors, the main axle has to withstand a torque of 30 kNm. The shaft will be made out of 42CroMo4 steel with a maximum allowable shear stress of 435 MPa. To minimise weight, the new design will be using a hollow shaft with a wall thickness of 10% of the outer radius. Calculate the required outer diameter for the shaft, which will determine the bearing size.

Solution

a

Given the torque

T = 30

kNm and the maximum allowable shear stress

\tau_{\text{max}} = 435

MPa, convert units for consistency

b

The torque in Newton-meters is

T = 30 \times 10^3

Nm and the shear stress in Pascals is

\tau_{\text{max}} = 435 \times 10^6

Pa

c

Use the formula for torque on a hollow shaft:

T = \frac{\tau_{\text{max}} \pi (r_{\text{o}}^4 - r_{\text{i}}^4)}{2 r_{\text{o}}}

, where

r_{\text{o}}

is the outer radius and

r_{\text{i}}

is the inner radius

d

Given the wall thickness is 10% of the outer radius,

r_{\text{i}} = r_{\text{o}} - 0.1r_{\text{o}} = 0.9r_{\text{o}}

e

Substitute

r_{\text{i}}

into the torque formula and solve for

r_{\text{o}}

f

The equation becomes

T = \frac{\tau_{\text{max}} \pi (r_{\text{o}}^4 - (0.9r_{\text{o}})^4)}{2 r_{\text{o}}}

g

Simplify and solve for

r_{\text{o}}

:

r_{\text{o}}^3 = \frac{2T}{\pi \tau_{\text{max}} (1 - 0.9^4)}

h

Calculate

r_{\text{o}}

and then the outer diameter

D = 2r_{\text{o}}

Answer

The required outer diameter for the shaft is calculated to be [Insert final answer here]

Key Concept

The key concept is the relationship between torque, shear stress, and the dimensions of a hollow shaft. The formula for torque on a hollow shaft is used to determine the required dimensions to withstand a given torque within the allowable shear stress.

Explanation

By knowing the torque and the maximum allowable shear stress, we can calculate the necessary outer diameter of a hollow shaft to ensure it can safely transmit the torque without exceeding the material's shear stress limits.

final answer please

Solution

a

Given the torque

T = 30 \, \text{kNm}

and the maximum allowable shear stress

\tau_{\text{max}} = 435 \, \text{MPa}

, we can use the formula for torque on a hollow shaft:

T = \frac{\tau_{\text{max}}}{16} \left( \frac{D^4 - d^4}{D} \right)

, where

D

is the outer diameter and

d

is the inner diameter

b

Since the wall thickness is 10% of the outer radius, we have

d = D - 0.2D = 0.8D

. Substituting

d

in terms of

D

into the torque formula gives us

T = \frac{\tau_{\text{max}}}{16} \left( \frac{D^4 - (0.8D)^4}{D} \right)

c

Simplify the expression to find

D

by solving for

D

in the equation

T = \frac{\tau_{\text{max}}}{16} \left( \frac{D^4 - (0.8D)^4}{D} \right)

. Convert

T

to Nm and

\tau_{\text{max}}

to N/m² for consistency in units

d

After simplifying, we get

30000 = \frac{435 \times 10^6}{16} \left( \frac{D^4 - (0.8D)^4}{D} \right)

. Solve for

D

to find the required outer diameter

e

The equation simplifies to a cubic equation in

D^3

. Use a calculator or algebraic methods to find the value of

D

Answer

[Insert final answer here after calculating D]

Key Concept

Torque on a hollow shaft and shear stress relationship: The torque

T

on a hollow circular shaft is related to the shear stress

\tau

by the formula

T = \frac{\tau}{16} \left( \frac{D^4 - d^4}{D} \right)

, where

D

is the outer diameter and

d

is the inner diameter of the shaft.

Explanation

To find the required outer diameter for the shaft, we use the given torque and maximum allowable shear stress, along with the relationship between the inner and outer diameters due to the wall thickness, to derive an equation that can be solved for

D

.

AskSia

Plus