Understanding Vehicle Suspension Systems

A vehicle suspension system is a critical component designed to absorb shocks and vibrations, providing a smooth ride and maintaining tire contact with the road surface. It typically consists of a spring and a damper (also known as a shock absorber). The spring stores energy and absorbs shocks, while the damper dissipates energy, controlling oscillations.

Key Parameters of the Suspension System

In this problem, we are given the following parameters for the vehicle suspension system:

Calculating the Damping Factor

The damping factor, often denoted by \(\zeta\) (zeta), is a dimensionless measure describing how oscillations in a system decay after a disturbance. It is the ratio of the actual damping constant (\(c\)) to the critical damping constant (\(c_c\)).

First, we need to calculate the natural frequency (\(\omega_n\)) of the undamped system:

\[\omega_n = \sqrt{\frac{k}{m}}\]

Substituting the given values:

\[\omega_n = \sqrt{\frac{3600 \, \text{N/m}}{50 \, \text{Kg}}} = \sqrt{72} \, \text{rad/s}\]

\[\omega_n \approx 8.485 \, \text{rad/s}\]

Next, we calculate the critical damping constant (\(c_c\)), which is the amount of damping required for the system to return to equilibrium as quickly as possible without oscillating:

\[c_c = 2 \sqrt{km} = 2m\omega_n\]

Using \(2m\omega_n\):

\[c_c = 2 \times 50 \, \text{Kg} \times 8.485 \, \text{rad/s} = 100 \times 8.485 \, \text{Ns/m} \approx 848.5 \, \text{Ns/m}\]

Alternatively, using \(2 \sqrt{km}\):

\[c_c = 2 \sqrt{3600 \, \text{N/m} \times 50 \, \text{Kg}} = 2 \sqrt{180000} \, \text{Ns/m} \approx 2 \times 424.26 \, \text{Ns/m} \approx 848.52 \, \text{Ns/m}\]

Now we can calculate the damping factor (\(\zeta\)):

\[\zeta = \frac{c}{c_c}\]

Using \(c = 400 \, \text{Ns/m}\) and \(c_c \approx 848.52 \, \text{Ns/m}\):

\[\zeta = \frac{400}{848.52} \approx 0.4714\]

So, the damping factor is approximately 0.471.

Calculating the Damped Natural Frequency

The damped natural frequency (\(\omega_d\) or \(f_d\)) is the frequency at which a damped system oscillates when disturbed from its equilibrium position. For an underdamped system (\(\zeta < 1\)), the damped natural frequency (\(\omega_d\)) in rad/s is related to the natural frequency (\(\omega_n\)) and the damping factor (\(\zeta\)) by the formula:

\[\omega_d = \omega_n \sqrt{1 - \zeta^2}\]

We found \(\omega_n \approx 8.485 \, \text{rad/s}\) and \(\zeta \approx 0.4714\). Substituting these values:

\[\omega_d = 8.485 \sqrt{1 - (0.4714)^2}\]

\[\omega_d = 8.485 \sqrt{1 - 0.2222}\]

\[\omega_d = 8.485 \sqrt{0.7778}\]

\[\omega_d = 8.485 \times 0.8819\]

\[\omega_d \approx 7.484 \, \text{rad/s}\]

The question asks for the frequency in Hertz (Hz). The relationship between angular frequency (\(\omega_d\) in rad/s) and frequency (\(f_d\) in Hz) is \(f_d = \frac{\omega_d}{2\pi}\).

\[f_d = \frac{7.484 \, \text{rad/s}}{2\pi}\]

\[f_d = \frac{7.484}{6.283} \, \text{Hz}\]

\[f_d \approx 1.191 \, \text{Hz}\]

So, the damped natural frequency is approximately 1.19 Hz.

Summary of Results

• Damping Factor (\(\zeta\)): Approximately 0.471
• Damped Natural Frequency (\(f_d\)): Approximately 1.19 Hz

These values represent how the suspension system will behave when subjected to disturbances. A damping factor between 0 and 1 indicates an underdamped system, meaning it will oscillate with decreasing amplitude over time at the damped natural frequency.

Based on our calculations, the damping factor is approximately 0.471 and the damped natural frequency is approximately 1.19 Hz. These values match the first option provided.