Question

An engineering company is designing and testing a car suspension system. The system has a conventional suspension design, consisting of a shock-absorber and spring at each wheel. The shock-absorber provides a damping effect that is proportional to the vertical speed (i.e. up and down motion) of the wheel, and the spring's resistance is proportional to the vertical displacement of the wheel.

The design team analyses the suspension system in two separate parts: Part 1 - dynamics of the spring-damper system; Part 2 - stress analysis of the coil spring.

 

PART 1

The company would like to investigate how the actual performance of the suspension system compares to that of theory. Dynamic theory states that the equation of motion of the wheel in the vertical direction can be expressed as,

   $LaTeX: m\frac{d^2y}{dt^2}=-c\frac{dy}{dt}-ky$ $m$ d 2 y d t 2 = − c d y d t − k y          or          $LaTeX: \frac{d^2y}{dt^2}+\frac{c}{m}\frac{dy}{dt}+\frac{k}{m}y=0$ $d$ 2 y d t 2 + c m d y d t + k m y = 0

where m is the mass of the wheel, k is the spring stiffness and c is the damping coefficient.

This is a second order differential equation, and if for example, the car hits a hole at t = 0, such that it is displaced from its equilibrium position with y = y0, and dy/dt = 0, it will have a solution of the form,

$LaTeX: y\left(t\right)=e^{-nt}\left(y_0\cos\left(pt\right)+y_0\left(\frac{n}{p}\right)\sin\left(pt\right)\right)$ $y$ ( t ) = e − n t ( y 0 cos ⁡ ( p t ) + y 0 ( n p ) sin ⁡ ( p t ) )

where         $LaTeX: p=\sqrt{\frac{k}{m}-\frac{c^2}{4m^2}}$ $p$ = k m − c 2 4 m 2          and          $LaTeX: n=\frac{c}{2m}$ $n$ = c 2 m         provided     $LaTeX: \frac{k}{m}>\frac{c^2}{4m^2}$ $k$ m > c 2 4 m 2

To analyse the actual performance of the design, the suspension system was built and tested with the following displacements of the wheel recorded during the time period of 2.6 and 3.1 seconds (this dataset is known to contain experimental error):

Time (s)	2.6	2.65	2.7	2.75	2.8	2.85	2.9	2.95	3	3.05	3.1
Displacement (m)	0.07	0.09	0.11	0.11	0.12	0.15	0.14	0.15	0.16	0.15	0.16

The design team are interested if the dataset can be characterised by expressions which are less complex than that of the above theory. 

 

PART 2

The design team performed stress analysis on the coil spring, and it was determined that the stress at a specific point could be characterised by:

Similar calculations were performed to calculate the maximum principal stress at 420 locations along a non-linear path through the coil spring. The following frequency distribution of the calculated maximum principal stresses was produced:

Max. Principal Stress (kPa)	4	4.5	5	5.5	6	6.5	7	7.5	8	8.5
Frequency	2	12	18	30	50	69	80	71	59	29

The design team is interested in the significance of these frequencies and therefore need to calculate the area under the graph which displays these results.

The design team believe any further calculations of principal stresses and determining the area under the graph will be time consuming (and potentially error prone), thus would like an automated method for performing this.

 

Submission

Create two separate MATLAB scripts for Part 1 and Part 2:

PART 1 - Create a MATLAB script which is capable of performing the following:

• Plotting the theoretical displacement of the wheel which shows the first 8 roots when the suspension system has the following specifications:

• mass acting on each wheel = 3.6 x 103 kg

• c = 1.5 x 103 Ns/m

• k = 1.5 x 104 N/m                        

• initial displacement = 0.3 m

• Calculating the time for the 3rd, 6th and 7th occasion when the wheel passes through the equilibrium position (i.e. the root);

• Calculating the value of the coefficient of multiple determination ($R$ 2 ) and the standard error (  $LaTeX: \sigma_E$ $\sigma$ E) when presuming the experimental dataset is characterised by a power equation ($y$ = a x b );

• Plotting a graph of the experimental dataset of the wheel displacement (displacement vs time) with a curve calculated using regression analysis when presuming the dataset is characterised by a power equation ($y$ = a x b);

• Calculating the displacement of the wheel at 2.8s when presuming the experimental dataset is characterised by a power equation ($y$ = a x b ).

 

PART 2 – Create a MATLAB script that is capable for performing the following:

• Calculating the maximum principal stress at the specific point shown;

• Plot the frequency distribution of the calculated maximum principal stresses and fit a natural spline through the dataset;

• Calculate the area under the natural spline (between the data points given).

 

Upload your two separate MATLAB script as .m files to Canvas. Please name the files in the following format: Part1_Forename_Surname.m and  Part2_Forename_Surname.m

Please note the following:

• The above can be achieved using various analytical techniques and user interface features.

• Consult the mark scheme when compiling to your script to ensure you include analytical features and user interface features that gain the most marks.

• Marks are only awarded for each section of the mark scheme provided the script runs successfully for that individual section. If a script/s fail at any point, then the marks associated with the mark scheme beyond the point of failure will be lost.