9.17  Lab instruction – Shaft and coupling

What is given

The motor and the requirements on the driveline are

The two misalignments are those of Figure 9.17.2 (a) and (b). The coupling must allow both at once.

The shaft is made of one of five materials, each with an allowable shear stress:

The couplings come from a practice catalogue of five RC couplings:

Its couplings fit shafts of 4, 5, 6 and 8 mm only, so the computed diameter is rounded up to one of these.

Procedure

1. Choose one of the five materials for the shaft.
2. Compute the running torque \(T\) of the motor, the design torque \(T_d\) and the smallest shaft diameter \(d_\min\) for your material.
3. Choose a standard diameter, 4, 5, 6 or 8 mm.
4. Write a Python program that reads the catalogue and keeps the couplings that
   1. carry the design torque \(T_d\), (b) allow both misalignments, angle and offset, and
   2. fit your shaft, with a largest bore at least your diameter.
5. Let the program print the lightest coupling that passes.
6. Parameter study: let the program plot how \(T_d\) changes when the motor speed varies from 5000 to 25 000 rpm, and mark your own operating point at 15 000 rpm.
7. Discussion: change the material in your program to the weakest (PLA) and then to the strongest (steel), and run it for both. How did the material change the size of the shaft and the coupling that was chosen in the end? What are the advantages and the drawbacks of a thick plastic shaft against a thin steel one in a racing car?

Analysis

The motor delivers the power \(P\) at the angular speed \(\omega = 2\pi n/60\), so its running torque is \(T = P/\omega\). A coupling and a shaft are sized for more than that: the service factorsv. servicefaktor \(K_s\) covers the shocks of the driveline, as for a coupling chosen from a catalogue in the couplings chapter,

\[ T = \frac{P}{\omega} = \frac{60\,P}{2\pi n}, \qquad T_d = K_s\,T \tag{9.17.1}\]

A solid round shaft in torsion has its largest shear stress at the surface, \(T/W\) with the torsional section modulus \(W = \pi d^3/16\) (8.13.5). Setting it equal to the allowable stress gives the smallest diameter,

\[ \tau_\max = \frac{16\,T_d}{\pi d^3} \le \tau_\text{allow} \quad\Rightarrow\quad d_\min = \left(\frac{16\,T_d}{\pi\,\tau_\text{allow}}\right)^{1/3} \tag{9.17.2}\]

The coupling is checked against the catalogue: its largest torque at least \(T_d\), its permitted angle and offset at least the required ones, and its largest bore at least the chosen diameter. The couplings chapter explains what each kind of misalignment does to a coupling that does not allow it.

We show the method with other numbers than those of the lab: a motor of 0.20 kW at 10 000 rpm, \(K_s = 1.5\) and a shaft of PETG.

P, n, K_s, tau_allow = 200.0, 10_000, 1.5, 18.0     # W, rpm, -, MPa (PETG)

T = P/(2*np.pi*n/60)*1e3                              # running torque, N mm
T_d = K_s*T                                           # design torque, N mm
d_min = (16*T_d/(np.pi*tau_allow))**(1/3)             # mm

The torques and the smallest diameter are

so this shaft is made 5 mm thick. The catalogue goes into a table, one row per coupling, and a condition on a column keeps the rows that meet it. The torque requirement alone keeps every coupling:

catalogue = pd.DataFrame({
    "type": ["rigid sleeve", "aluminium clamp", "jaw, plastic", "jaw, aluminium", "Oldham"],
    "T_max": [1.2, 1.5, 1.0, 2.5, 4.0],           # N m
    "angle": [0.0, 1.5, 2.0, 1.5, 1.0],           # degrees
    "offset": [0.00, 0.20, 0.40, 0.35, 1.00],     # mm
    "bore": [4, 5, 6, 6, 8],                      # largest bore, mm
    "mass": [4, 8, 5, 12, 22]},                   # g
    index=list("ABCDE"))

catalogue[catalogue["T_max"] >= T_d/1e3]

Your program adds the conditions on the misalignment and the bore, joined with &, and picks the lightest of the couplings that remain, for instance with nsmallest(1, "mass"). For the parameter study, \(T_d\) from 9.17.1 falls as \(1/n\): the same power at a lower speed needs a larger torque, and so a thicker shaft.

Report

Hand in on Canvas, as two files:

• your Python code, as a .py file or a Jupyter notebook, and
• a report of at most one A4 page as a PDF, with text, a table and a graph only.

The report holds

1. a table of your computed values: \(T\), \(T_d\), the theoretical diameter and the chosen standard diameter,
2. the name of the coupling your program chose, with a short reason,
3. the graph of the speed study, and
4. your answer to the discussion question, step 7 of the procedure.

Questions

Answer with the help of your program and the shaft–hub chapter.

1. Which requirement removes the most couplings from the catalogue: the torque, the misalignment or the bore? Does that change with the material of the shaft?
2. For PLA and for steel, how much more torque than \(T_d\) can your shaft carry at the standard diameter you chose? Which margin is larger, and why?
3. In the speed study, at which speed would your chosen coupling no longer carry \(T_d\)?