## The company would like to investigate how the actual performance of the suspension system compares to that of theory.

### engineering

##### Description

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,

m d 2 y d t 2 = − c d y d t − k y          or         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,

y ( t ) = e − n t ( y 0 cos ⁡ ( p t ) + y 0 ( n p ) sin ⁡ ( p t ) )

where        p = k m − c 2 4 m 2          and         n = c 2 m         provided    k m > c 2 4 m 2

 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

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):

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