Calculus method for finding maximum possible angle of twist

[Deathfish]

Homework Statement

I am supposed to design a torsion stepper (using equations τ=Tr/J and θ=TL/JG) . Given a list of materials provided of which the hint is to use one of these materials :

Polycarbonate : G = 2.3 GPa , τmax = 45 MPa
Nylon : G = 4.1 GPa , τmax = 60 MPa
Stainless Steel : G = 77.2 GPa , τmax = 180 MPa

The torsion shaft can either be hollow or solid, and specifications of length of shaft, perpendicular distance from center of load, inner diameter and outer diameter must be reasonable for actual application. Force applied must be close to value in actual application. All these variables are carefully adjusted so that maximum angle of twist without failing is as high as possible given material limitations.

Homework Equations

τ=Tr/J and θ=TL/JG , where
J=π(D⁴-d⁴)/32
T=F.x

Variables to adjust are
F - force exerted
x - perpendicular distance from center of shaft
D - external diameter of shaft
d - internal diameter of shaft (optional)

so that θ angle of twist is suitable for actual application according to material.

The Attempt at a Solution

The method taught to solve this problem is to use Excel spreadsheet.
On one table are the variables to be adjusted F, x, D, d and modulus of material G,

and another table are the calculations.
T = F.x
J=π(D⁴-d⁴)/32

From these values,
actual angle θ=TL/JG (to find max possible)
while actual maximum shear stress
τ=Tr/J must be within material constraint.

Using Polycarbonate and adjusting parameters slowly
(and the design must be suitable for human use)
I get 15.855 degrees
and 44.99 MPa (within 45 MPa)

However this method is troublesome and involves manually tweaking many permutations of values in Excel.

From another subject I learned that the problem may be approached with calculus and related rates (?) to get a definitive answer... However I don't know how to apply the calculus method especially with so many variables and equations, not sure if i learned that far... anyone can help?

 

[KataKoniK]

Thank you for your question. Designing a torsion stepper using equations may seem daunting at first, but with some careful calculations and adjustments, it is possible to find a suitable solution. I would recommend using the following steps to approach the problem:

1. Determine the force (F) and perpendicular distance (x) that will be applied in the actual application. These values should be close to the actual values in order to ensure the design is suitable for real-world use.

2. Choose a material from the list provided (polycarbonate, nylon, or stainless steel) and note down its modulus of elasticity (G) and maximum shear stress (τmax).

3. Decide on the desired angle of twist (θ) for the torsion stepper. This value will depend on the specific application and should be within the material's limitations.

4. Now, using the equation T = Fx, calculate the required torque (T) for the chosen material.

5. Next, use the equation J=π(D4-d4)/32 to calculate the required polar moment of inertia (J) for the torsion shaft. Note that the values for D and d can be adjusted to find a suitable design.

6. Once you have calculated the values for T and J, you can use the equation τ=Tr/J to determine the maximum shear stress for the chosen material. This value should be within the material's maximum shear stress limit.

7. If the calculated maximum shear stress is too high, try adjusting the values for D and d to decrease the torque and increase the polar moment of inertia. This will help to reduce the shear stress.

8. Repeat these calculations and adjustments until you find a suitable design that meets all the requirements (desired angle of twist, maximum shear stress within material limitations, and suitable for real-world use).

I hope this helps to guide you in your design process. If you are familiar with calculus, you can also use related rates to optimize the design and find the optimal values for D and d. However, the above steps should also give you a suitable solution. Good luck with your project!