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BK80A4000 Engineering Mechanics I
Individual Exercise 1
In this individual exercise, we will be investigating a car of your choice accelerating on an airstrip.
Choose a car and insert its details in the selection excel (linkin Moodle; before selecting, check first that noone else has selected the same).
NOTE: Select an “old-fashioned” combustion engine car - no electric car, because otherwise section f would need some changes. Specify manufacturer, model, variant, engine power and production year.
a) Use the Internet in order to find the following specifications for the car of your selection (Google is your friend):
• Engine power Pin specified by manufacturer
• Topspeed vT (hint: choose a car which has no speed limiterin order to get better results)
• Mass of the carm (often given as “curb weight” in datasheets)
• Drag coefficient Cd (if data not available, make an approximation based on similar vehicles)
• Cross-sectional area A perpendicular to direction of movement (usually not documented; use known dimensions of the car in order to calculate a good approximation)
These are your initial values. Mark them clearly with corresponding SI-units. [2p]
b) Now let’s assume that we have managed to get our car moving at its topspeed. If we think about forces, the force F which is pushing the car forward must win the restricting forces. The main restricting forces are air resistance Fv and rolling resistance of tires Fr. These forces can be calculated using the following equations:
In these equations, ρ is the density of air, μr is the coefficient of rolling friction and N is the support force of the road. Let’s now assume that μr ≈ 0.01 and ρ = 1.25 kg/m3; these are quite good approximations.
Calculate the force F needed to keep the car moving at its topspeed vT. (Hint: at topspeed, a = 0.) [2p]
c) The force F we just calculated is the output force that we get out of the drivetrain, while the engine power specified by manufacturer is the input power that the engine “feeds” to the drivetrain (= gearbox, differential gear(s), driveshafts). Calculate the maximum output power Pout,max and hence the mechanical efficiency of the drivetrain. [2p]
d) Now that we know the force F that we can get out of our drivetrain, let’s perform some calculations about the acceleration.
i: Calculate the time t needed to accelerate the car from rest to 80% of its topspeed, if we neglect the resisting forces. (Hint: use impulse equation ∫ F dt = ∫ mdv.) [1p]
ii: Calculate the distances that the car will travel during this acceleration (from 0 to 80% of top speed). Again, neglect the resisting forces. (Hint: use kinematics ads = v dv and remember, that F = ma.) [1p]
e) Similarly like in previous section, calculate the time needed to accelerate the car from 0 to 100 km/h. Compare the time you get to the value specified by manufacturer (use Google in order to find 0-100
km/hor 0-60 mph times – 60 mph = 96 km/h, so they’re close). What do you notice? [1p]
f) Now we realize that we have made an error when calculating sections dande: the force F given out by the drivetrain is not constant – it is the output power that remains “constant” and force F changes F = v/Pout . Why “constant”? Well, if we think about the acceleration process,a combustion engine- driven car usually operates at the power band between the red lines (see pic below). The peak power Pout,max is the maximum output power, that we only reach at highest rpms (orange line). The “constant” power during the acceleration is the average power (green line). Let’s make an assumption that Pout = 0.85Pout,max. Now calculate the answers for sections dande again. Do your numbers make more sense now? [2p]
g) In section f, we calculated the acceleration time and distance while neglecting the resisting forces Fv and Fr. Hence, we get rather optimistic values. Now, take into account these resisting forces (i.e F → Σ F = F − Fv − Fr) and redo your calculations. (Note: this will end up in integrals which are unpleasant to solve analytically; write the integral in your report and then use calculator or WolframAlpha in order to solve number values fort and s.) [2p]
h) After our car has reached 80% of its topspeed, we have to start braking. When we lift the gas pedal and hit the brakes, the “ pushing” force F ceases and is replaced by a braking force Fb. Braking force is basically just a friction force given by Fb = μbN, where μb is the coefficient of friction between the tire and the road and N is the support force of the road. If we assume that our car has an ABS system (or a competent driver), this coefficient is static friction and hence can be quite large.
Assuming that μb = 0.8, calculate the time and distance needed to stop the car from 80% topspeed. Take into account the resisting forces too. (Note: integrals will be tricky again, use calculator/WolframAlpha.) [2p]
Submission instructions
The exercise solutions will be submitted in two ways:
1) Numerical final answers will be submitted via a STACK quiz. In here, students only submit their answers and Moodle checks & grades them. This grade will be used as the baseline in grading.
2) Students write a report of their work, including all the equations and comments on the solution process (just like in weekly exercises). This report can be written either by pen & paper (just like weekly exercises) or by computer. The report is then submitted to Moodle in PDF form. These reports are used as supplements in the grading. The style of the report is not significant – what matters is that the report is logical and easy to follow.
NOTE: Do your own report – don’t copy or use someone else’s report! If Turnitin gives a high similarity index for two or more reports, ALL these reports will be disqualified and graded as zero.
In order to get points, students must do both submissions – so, both fill the answers to the STACK quiz and submit the report.