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.
To design the suspension system for the given half-car model, we need to determine the spring constants and damping constants. The first step is to calculate the natural frequencies for the predominantly bouncing and pitching modes.
1. Natural Frequency for Bounce Motion:
The natural frequency for the bounce motion can be calculated using the formula:
ωn_bounce = √(k_bounce / m)
where ωn_bounce is the natural frequency for bounce motion, k_bounce is the spring constant, and m is the mass of the half-car.
We want the natural frequency to be between 1.5 Hz and 2.2 Hz. Convert these values to rad/s by multiplying by 2π:
ωn_bounce_min = 1.5 Hz * 2π ≈ 9.42 rad/s
ωn_bounce_max = 2.2 Hz * 2π ≈ 13.82 rad/s
2. Natural Frequency for Pitch Motion:
The natural frequency for the pitch motion can be calculated using the formula:
ωn_pitch = √(k_pitch / I)
where ωn_pitch is the natural frequency for pitch motion, k_pitch is the spring constant, and I is the moment of inertia about the center of mass.
We want the natural frequency to be between 2.5 Hz and 3.0 Hz. Convert these values to rad/s:
ωn_pitch_min = 2.5 Hz * 2π ≈ 15.71 rad/s
ωn_pitch_max = 3.0 Hz * 2π ≈ 18.85 rad/s
Now, we can calculate the spring constants for bounce and pitch motions using the known natural frequencies and the formulas mentioned earlier:
1. Spring Constant for Bounce Motion (k_bounce):
From the natural frequency formula for bounce motion:
k_bounce = (ωn_bounce)^2 * m
Substituting the given values:
k_bounce_min = (9.42 rad/s)^2 * 0.8 metric ton
k_bounce_max = (13.82 rad/s)^2 * 0.8 metric ton
2. Spring Constant for Pitch Motion (k_pitch):
From the natural frequency formula for pitch motion:
k_pitch = (ωn_pitch)^2 * I
Substituting the given values:
k_pitch_min = (15.71 rad/s)^2 * 260 kg-m^2
k_pitch_max = (18.85 rad/s)^2 * 260 kg-m^2
Next, let's calculate the damping ratios for each quarter car system using the given formulas:
damping ratio rear quarter car (ζ1):
ζ1 = b1 / (2m * ωn_bounce)
damping ratio front quarter car (ζ2):
ζ2 = b2 / (2m * ωn_pitch)
We want the damping ratios to be between 0.5 and 0.6.
Now that we have all the necessary formulas and values, you can substitute the given quantities into the equations to calculate the spring constants (k_bounce and k_pitch) and the damping ratios (ζ1 and ζ2) for the suspension system.