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EG-264 CAE MATLAB Assignment 2019/2020
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Question 1:
Figure 1 shows the speed (mph) against time(s) graph of a vehicle accelerating from a standing start. Over 8
seconds the car accelerates to nearly 150mph.
Figure 1: Speed (mph) vs. Time(s) graph of a vehicle in motion.
Equation (1) reproduces the speed from time inputs as in figure 1.
Speed(t) = 0.0041(t6) – 0.1383(t5) + 1.6963(t4) – 8.915(t3) +13.961 (t2) + 40.96(t) (Eqn. 1)
To determine the distance travelled by the vehicle, numerical integration can be used on the reproduced speed
vs. time data, calculating the area under the curve.
Any of the three methods of numerical integration taught during the module (Composite Mid-Point,
Trapezoidal or Simpsons Rule) can be used to determine the distance travelled of the vehicle represented in
Figure 1/ Equation 1. Clearly state whichever method you are using, but you must obtain an approximation
of the distance travelled by the vehicle in SI units, with a relative error of less than 0.00002%.
During the numerical integration calculations, if the relative error is not reached, double the number of
separations used over the timespan in the calculations for the following calculation.
(i) In the command window, display the integral value calculated for distance, the number of
sections used in the numerical integration, and the relative error produced for each looped
calculation using ‘fprintf’ and associated commands.
(ii) Produce a single figure with two subplots, (1) showing the speed (m/s) vs. time (t) of the speed
equation in one plot at a reasonable accuracy, and (2) a cumulative distance (m) graph of the
vehicle over time(s) in the second subplot.
(iii) Produce a figure showing the total distance calculated against the number of separations used in
each numerical integration calculation; use a logarithmic x-axis scale on the resulting plot.
[25 Marks]
EG-264 CAE MATLAB Assignment 2019/2020
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Question 2:
An underdamped system is excited, with an initial velocity 𝑣0, which produces a vibration of the system. The
displacement of the system over time (x(t)) can be calculated using Equation 4, when combined with
equations 2, 3 and the parameter values detailed in Table 1.
Table 1: Parameter definitions, values and units of measure
Definition Parameter Value Units
Stiffness 𝑘 997.584 N/m
Mass 𝑚 10.692 kg
Initial Displacement 𝑥0 0.026844 m
Initial Velocity 𝑣0 Unknown m/s
Time 𝑡 0.735 s
Displacement 𝑥 0.0852 m
Damping coefficient c 36.84 kg/s
Natural Frequency 𝜔𝑛 𝜔𝑛 = √
Damped Natural Frequency 𝜔𝑑 𝜔𝑑 = 𝜔𝑛√1 − 𝜁2 rad/s
Critical Damping Coefficient 𝑐𝑐𝑟𝑖𝑡 𝑐𝑐𝑟𝑖𝑡 = 2√𝑘 ∗ 𝑚 kg/s
Damping Ratio 𝜁 𝜁 = 𝑐/𝑐𝑐𝑟𝑖𝑡 -
The initial velocity 𝑣0 however is unknown.
(i) Use the bisection method to determine an approximation of the value of 𝑣0, with an
absolute error of less than 1 ∗ 10−6
; when the displacement x = 0.0852m, at time t =
0.735s, limiting the useable values of 𝑣0 between: 0 < 𝑣0 < 5. Through each loop,
display the resulting value of 𝑣0, and the absolute error obtained.
(ii) Produce a plot of the displacement x(t), between 0 < 𝑡 < 4𝑠, using the initial velocity
𝑣0 calculated in part (i), and highlight the vibration displacement (x(t)), at the time (s)
stated in table 1.
[25 MARKS]