Machine Design 5

Laboratory report

Experiment 3: Screw mechanisms and joints.

The purpose of the experiment is to determine stresses in screw mechanisms, bolts and fasteners and the efficiency of the screw mechanism.

1.1 Stand diagram:

1. bolt

2. nut

3. thrust washer

4. compound strain gauge transducer sleeve

5. strain gauge transducer sleeve

1.2. Measurement principles and results calculations.

The bolt built into the stand has an M30 thread. According to ISO standard the dimensions of thread are:

α = 60 deg

pitch: P=3,5 mm

thread major diameter: d=30mm

According to ISO standard:

thread minor diameter: d₃=25.71 mm

nut minor diameter*: d₁=26.5 mm

The dimensions of the bolt:

d_W = 20 mm

d_Z = 25 mm

As the wrench is being turned the torque moment gradually increases. The strain gauges mounted on element 4 (compound strain gauge transducer sleeve) are used to measure the axial stress Q as well as the torque of the thread resistance M_G, while the strain gauge mounted on element 5 measures the friction torque of the feet of the bolt M_P. The fourth measured quantity is the torque moment applied using the key M_KL.

These 4 measurements are stored in a PC type computer using an analog/digital converter.

The μ’ (apparent friction coefficient of the thread surface) and μ_P (coefficient of friction on contact plane at the base of the bolt) are calculated using relations derived from following equations (for case when bolt is turned clockwise):

Resulting in

$$\mathbf{\mu' =}\frac{\left( \frac{2\mathbf{M}\mathrm{G}}{\mathbf{d}\mathrm{s}\mathbf{Q}} \right)\mathbf{-}\tan\mathbf{\gamma}}{\left( \frac{2\mathbf{M}\mathrm{G}}{\mathbf{d}\mathrm{s}\mathbf{Q}} \right)\tan\mathbf{\gamma + 1}}$$

$$\mathbf{\mu}\mathrm{P}\mathbf{=}\frac{\mathrm{2}\mathrm{M}\mathrm{P}}{\mathrm{d}\mathrm{m}\mathrm{Q}}$$

where :

- where ρ’ is the apparent angle of friction

d_m = ½(d_w+d_z) - average friction diameter of bolt base

d_s = ½ (d+d₁) - average friction diameter of thread surface

γ = atan(P/πd_s) - average helix angle

(note: R. L. Norton in his Machine Design proposed a different formulae d_s=d-0.649519P)

The PC program available at the stand does not calculate μ_G, however this calculation could also be automated.

The program calculates the thread efficiency using following formulae (derived in experiment instructions):

$$\eta\mathrm{G =}\frac{\mathrm{L}\mathrm{u}}{\mathrm{L}\mathrm{w}}\mathrm{=}\frac{\mathrm{\text{QP}}}{\mathrm{2}\mathrm{\pi}\mathrm{M}\mathrm{G}}\mathrm{=}\frac{\mathrm{\tan}\mathrm{\gamma}}{\mathrm{\tan}\left( \mathrm{\gamma}\mathrm{+}\mathrm{\rho}^{\mathrm{'}} \right)}$$

1.3. Main stand components and stress distribution diagrams:

1.4. Measurement errors:

Due to an automated registration of data the errors of measurements in this experiment.

The accuracy of measurement of voltage is estimated at $\frac{U_{\text{CZ}}}{U} = 0.002$, resulting in following measurement errors:

$$\frac{\mathbf{Q}}{\mathbf{Q}}\mathbf{=}\frac{U_{Q}}{U_{Q}} = 0.022$$

$$\frac{\mathbf{M}_{\mathbf{G}}}{\mathbf{M}_{\mathbf{G}}}\mathbf{=}\frac{U_{G}}{U_{G}} = 0.022$$

$$\frac{\mathbf{M}_{\mathbf{P}}}{\mathbf{M}_{\mathbf{P}}}\mathbf{=}\frac{U_{P}}{U_{P}} = 0.042$$

The values of errors of calculated parameters therefore are:

$$\mu_{G} = \left| \frac{\partial\mu_{G}}{\partial\mathbf{Q}} \right|\mathbf{Q} + \left| \frac{\partial\mu_{G}}{\partial\mathbf{M}_{\mathbf{G}}} \right|\mathbf{}\mathbf{M}_{\mathbf{G}}\mathbf{=}\cos\frac{\propto}{2}\left( \left| \frac{\partial\mu^{'}}{\partial\mathbf{Q}} \right|\mathbf{Q} + \left| \frac{\partial\mu^{'}}{\partial\mathbf{M}_{\mathbf{G}}} \right|\mathbf{}\mathbf{M}_{\mathbf{G}} \right) = = cos\frac{\propto}{2}\frac{\left( \mu^{'} + tan\gamma \right)\left( 1 - \mu^{'}\tan\gamma \right)}{1 + \tan^{2}\gamma}\left( \frac{\mathbf{Q}}{\mathbf{Q}} + \frac{\mathbf{}\mathbf{M}_{\mathbf{G}}}{\mathbf{M}_{\mathbf{G}}} \right) = = cos\frac{\propto}{2}\frac{\left( \mu^{'} + tan\gamma \right)\left( 1 - \mu^{'}\tan\gamma \right)}{1 + \tan^{2}\gamma}0.044$$

$$\mu_{P} = \left| \frac{\partial\mu_{P}}{\partial\mathbf{Q}} \right|\mathbf{Q} + \left| \frac{\partial\mu_{P}}{\partial\mathbf{M}_{\mathbf{P}}} \right|\mathbf{}\mathbf{M}_{\mathbf{P}}\mathbf{=}\left| \frac{2\mathbf{M}_{\mathbf{P}}}{d_{m}\mathbf{Q}} \right|\left( \frac{\mathbf{Q}}{\mathbf{Q}} + \frac{\mathbf{}\mathbf{M}_{\mathbf{P}}}{\mathbf{M}_{\mathbf{P}}} \right)\mathbf{=}0.064\mu_{P}$$

Errors for μ_G, μ_P and the values of μ_G for 5 selected points re are:

1.5. Closing conclusions:

The error interval boarders in μ_P graph are not aligned correctly. This is due to the fact I was reading values from unfiltered graph. I expect the same kind of error in case of hand drawn μ_G graph, however smaller since the source graph μ’ is somewhat smoother.

Filtering the graphs cannot be done manually and it would be good to introduce it as a function of used program. Another possible improvement would be to draw μ_G automatically.

The graphs show the friction coefficients first decreasing but above some Q being more or less constant. The friction coefficient at the feet of the bolt being larger than the apparent and real friction coefficients of the thread surface.

The efficiency of mechanism seems to be constant for changing values of Q.

However the broad error interval does not allow for certainty.

The best observed for this experiment is the linear relation between torque and axial force in the bolt. This is also what theory suggests.