The suspension system of a 2000 kg automobile “sags” L0 cm when the chassis is placed on it. Also, the oscillation amplitude decreases by 50% each cycle. 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 each wheel supports 500 kg.
The correct answer and explanation is:
Solution:
(a) Spring Constant kk:
Each wheel supports a mass of m=2000 kg4=500 kgm = \frac{2000 \text{ kg}}{4} = 500 \text{ kg}
The weight of this mass is F=mg=(500)(9.81)=4905 NF = mg = (500)(9.81) = 4905 \text{ N}
From Hooke’s Law, the force is related to the displacement L0L_0 by: F=kL0F = kL_0
Thus, the spring constant is: k=FL0=4905L0k = \frac{F}{L_0} = \frac{4905}{L_0}
(b) Damping Constant bb:
The amplitude of oscillation decreases by 50% per cycle, meaning the damping ratio is significant. The logarithmic decrement δ\delta is given by: δ=lnx1x2=ln2=0.693\delta = \ln \frac{x_1}{x_2} = \ln 2 = 0.693
The damping ratio ζ\zeta is related to δ\delta by: ζ=δ4π2+δ2\zeta = \frac{\delta}{\sqrt{4\pi^2 + \delta^2}}
Substituting δ=0.693\delta = 0.693, ζ=0.6934π2+0.6932≈0.110\zeta = \frac{0.693}{\sqrt{4\pi^2 + 0.693^2}} \approx 0.110
The natural frequency of the undamped system is: ω0=km\omega_0 = \sqrt{\frac{k}{m}}
The damping coefficient is: b=2mζω0b = 2m\zeta\omega_0
Substituting values, b=2(500)(0.110)k500b = 2(500)(0.110) \sqrt{\frac{k}{500}}
Explanation:
The suspension system in an automobile comprises a spring and a shock absorber. The spring supports the weight of the vehicle, while the shock absorber dissipates energy to reduce oscillations. By knowing the static sag (L0L_0), we estimate the spring constant. The damping constant is derived from the decay of oscillation amplitude. Proper damping ensures the car does not bounce excessively, improving stability and comfort.
Now, I’ll generate an image illustrating the suspension system and key parameters.
Here is the labeled diagram of a car suspension system, showing the key parameters like mass (mm), spring constant (kk), damping coefficient (bb), displacement (L0L_0), and oscillation decay. Let me know if you need any modifications or further explanations!
