A half-car model of an automobile suspension system is shown below. You are to design its suspension systems (values of spring constants and damping constants). Generally, for ride comfort, it is desired to have the bounce motion at low frequency and the pitch motion at
relatively higher frequency for handling and control.Assuming the mass of the half-car is 0.8 metric ton, the moment of inertia about
the center of mass being 260 kg-m2, and the wheel base of 2 meters, design the suspension system such that the predominantly bouncing mode (or approximate bouncing) occurs at a frequency between 1.5 Hz and 2.2 Hz and the predominantly pitching mode (or approximate pitching) between 2.5 to 3.0 Hz. The center of mass of the car does not have to be at the geometrical center. Also, for each quarter car system, the damping ratio should be between 0.5 and 0.6. The damping ratio for each quarter car is defined as :
£ii=bi/2m wni where i=1 is for the
rear quarter car and i=2 is for the front quarter car, and £sn is the natural frequency for the quarter car.
Estimate and discuss the maximum bouncing and pitching generated by your
design when the car travels at 40 km/hr over a rugged terrain roughly modeled by a sine wave
with 0.10 meter amplitude and 4.0 meter period. (This one does not have to be an accurate
calculation. State all your assumptions for this estimation.) Be as detailed in your discussion as
possible!
To design the suspension system with the given specifications, we need to calculate the spring constants and damping constants for the rear and front quarter car systems separately. Let's go step by step:
1. Calculate the natural frequencies for bounce and pitch motions:
The natural frequencies for the quarter car systems can be calculated as follows:
Bounce natural frequency:
wn_bounce = sqrt(k_rear/m) + sqrt(k_front/m)
Pitch natural frequency:
wn_pitch = sqrt((k_rear + k_front)/(2*I))
where k_rear and k_front are the spring constants for the rear and front quarter cars, m is the mass of the half-car, and I is the moment of inertia.
2. Set the natural frequencies within the desired ranges:
Based on the given specifications, the bounce natural frequency should be between 1.5 Hz and 2.2 Hz, and the pitch natural frequency should be between 2.5 Hz and 3.0 Hz.
3. Calculate the damping ratios for each quarter car system:
The damping ratios for the quarter car systems can be calculated using the formula:
ζ_i = (b_i / (2 * m * wn_i))
where ζ_i is the damping ratio, b_i is the damping constant for the ith quarter car, m is the mass of the half-car, and wn_i is the natural frequency for the ith quarter car.
4. Set the damping ratios within the desired range:
According to the given specifications, the damping ratio should be between 0.5 and 0.6 for each quarter car system.
5. Solve the equations and find the values:
Using the formulas and the given values for mass, moment of inertia, and wheelbase, you can solve the equations to find the appropriate values for the spring constants (k_rear and k_front) and damping constants (b_rear and b_front).
Note: There might be multiple valid combinations of spring and damping constants that satisfy the given specifications. You can use numerical methods or trial and error to find the most suitable values.
By following these steps and performing the necessary calculations, you can design the suspension system with the desired characteristics.