You are riding in an automobile of mass 3000 kg. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags 15 cm when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by 50% during one complete oscillation. Estimate the values of (a) the spring constant k and (b) the damping constant b for the spring and shock absorber system of one wheel, assuming that each wheel supports 750 kg.
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(a)Mass of the automobile ( m) = 3000 kg
Displacement in the suspension system x = 15 cm = 0.15 m
There are 4 spring in parallel combination so,
Keq = 4K
F = - 4Kx
But a/c to Newton's 2nd law,
F = mg
K = mg /4x
= 3000 × 9.8/4 × 0.15
= 4.9 × 10⁴ N/m ≈ 5 × 10⁴ N/m
(b) mass of each wheel = 3000/4 = 750 kg
For damping factor b , the equation of displacement is
x = xo.e^(-bt/2m)
as x = xo/2
xo/2 = xoe^(-bt/2m)
b = 2m.ln2/t ------(1)
The time taken in 50% damping = one time period = T
T = 2π√(m/4K)
= 2π√(3000/4 × 5 × 10⁴)
= 0.769 put t = T = 0.769 in eqn (1)
b = 2 × 750 × ln2/0.769
= 1351.58 kg/s
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