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###### 日期：2019-11-03 09:29

EG-264 CAE MATLAB Assignment 2019/2020

Page 2 of 3

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

Page 3 of 3

Question 2:

An underdamped system is excited, with an initial velocity ??0, 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 ?? 997.584 N/m

Mass ?? 10.692 kg

Initial Displacement ??0 0.026844 m

Initial Velocity ??0 Unknown m/s

Time ?? 0.735 s

Displacement ?? 0.0852 m

Damping coefficient c 36.84 kg/s

Natural Frequency ???? ???? = √

Damped Natural Frequency ???? ???? = ????√1 ? ??2 rad/s

Critical Damping Coefficient ?????????? ?????????? = 2√?? ? ?? kg/s

Damping Ratio ?? ?? = ??/?????????? -

The initial velocity ??0 however is unknown.

(i) Use the bisection method to determine an approximation of the value of ??0, 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 ??0 between: 0 < ??0 < 5. Through each loop,

display the resulting value of ??0, and the absolute error obtained.

(ii) Produce a plot of the displacement x(t), between 0 < ?? < 4??, using the initial velocity

??0 calculated in part (i), and highlight the vibration displacement (x(t)), at the time (s)

stated in table 1.

[25 MARKS]