Construct the mathematical function that describes the movement of the piston.

1.Relevant calculations and construction of the algebraic model. Parameter adjustment

If the distance d that the piston travels inside the piston is 1 m, the amplitude of the wave that describes the sinusoidal motion is 0.5 m. On the other hand, the frequency ( fr ) with which the wheel that moves the connecting rod moves is 30 Hz. With this data you can calculate the angular frequency

= 2 fr = 2*30 Hz = 188.50 rad/s

and B = A = A2fr = 0.5*2*30 Hz = 94.25 rad/s. If the speed of the train were V _T = 30 km/h = 8.33 m/s, the algebraic model would be the following:

v = -B sen (t) = -94.25 sen (188.50 t) + 8.33 m/s

2. Interpretation of graphics

Build the velocity graph with the function obtained, for t from 0 to 0.2 s, from 0.001 to 0.001 s and save it with the name piston.xls .

3.Modifying parameters

Suppose you now take the following values:

Parameters	Values
A (m)	1.7
fr (Hz)	30
In T1 (m/s)	14

Write the new function and make the graph on sheet 2 of your file. Compare your new graph with the old one. What difference do you notice between them? What does this frequency value mean compared to the previous model?

4.Predictions

With these calculations made, predict what will happen if you keep the values in point 3 and modify:

Parameters	Values
VT2 (m/s)	20

Write the function, graph it on sheet 3 of your file and answer right there.

5. Scope and limitation of the model

For this model, the speed of the piston and the locomotive is considered constant. Assume there is no wear on the plunger inside the piston (no friction). If you take a gasoline engine as an example, the number of turns will be much greater and the speed of both the piston and the car could be greater.

6. Final challenge

What would happen if you varied the frequency in your piston model from point 3 of the problem. How would you expect the graph to be modified? Perform the relevant calculations, write the resulting function and graph it on sheet 4.

Parameters	Values
fr (Hz)	32

Save your file with the name Pistn.lxls

Carry out your activity in spreadsheets (Excel)