A vehicle suspension system consists of a leaf spring and a damper. The stiffness of the leaf spring is 3.6 kN/m and damping constant of the damper is 400 Ns/m. If the mass is 50 kg, then the damping factor and damped natural frequency respectively are
0.471 and 1.19 Hz
Vehicle Suspension Analysis: Damping Factor & Frequency Calculation
This solution details the process of calculating the damping factor and the damped natural frequency for a given vehicle suspension system. We will use the provided values for mass, stiffness, and damping constant.
Vehicle Suspension Parameters Explained
The vehicle suspension system is modeled as a mass-spring-damper system. The essential components and their properties are:
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Mass | $m$ | 50 | kg |
| Stiffness (Leaf Spring) | $k$ | 3.6 | kN/m |
| Damping Constant (Damper) | $c$ | 400 | Ns/m |
It is important to convert the stiffness value to standard SI units (N/m) for calculations: $k = 3.6 \, \text{kN/m} = 3600 \, \text{N/m}$.
Suspension System Formulas Overview
The core calculations rely on fundamental concepts in vibration analysis:
- Natural Angular Frequency ($\omega_n$): This represents the oscillation frequency of the system if there were no damping. It is calculated using the formula: $$ \omega_n = \sqrt{\frac{k}{m}} $$
- Damping Factor ($\zeta$): Also referred to as the damping ratio, this dimensionless value quantifies the level of damping relative to critical damping. The formula involves the damping constant ($c$) and the critical damping coefficient ($c_c$): $$ \zeta = \frac{c}{c_c} $$ The critical damping coefficient ($c_c$) is determined by: $$ c_c = 2\sqrt{mk} $$ Thus, the damping factor can be expressed as: $$ \zeta = \frac{c}{2\sqrt{mk}} $$
- Damped Natural Frequency ($\omega_d$): For systems with damping (where $\zeta < 1$), this is the actual frequency of oscillation. It is calculated as: $$ \omega_d = \omega_n \sqrt{1 - \zeta^2} $$
- Frequency in Hertz ($f_d$): To express the damped frequency in cycles per second (Hertz), convert the angular frequency (radians per second) using: $$ f_d = \frac{\omega_d}{2\pi} $$
Calculating Damping Factor and Damped Frequency
We will now compute the required values step-by-step:
1. Calculate Natural Angular Frequency ($\omega_n$)
Using the mass ($m = 50$ kg) and stiffness ($k = 3600$ N/m):
$$ \omega_n = \sqrt{\frac{3600 \, \text{N/m}}{50 \, \text{kg}}} $$ $$ \omega_n = \sqrt{72 \, \text{s}^{-2}} $$ $$ \omega_n \approx 8.485 \, \text{rad/s} $$2. Calculate Damping Factor ($\zeta$)
First, determine the critical damping coefficient ($c_c$):
$$ c_c = 2\sqrt{mk} = 2\sqrt{50 \, \text{kg} \times 3600 \, \text{N/m}} $$ $$ c_c = 2\sqrt{180000 \, \text{kg}^2/\text{s}^2} $$ $$ c_c = 2 \times 424.26 \, \text{kg/s} $$ $$ c_c \approx 848.53 \, \text{Ns/m} $$Next, use the given damping constant ($c = 400$ Ns/m) to calculate the damping factor ($\zeta$):
$$ \zeta = \frac{c}{c_c} = \frac{400 \, \text{Ns/m}}{848.53 \, \text{Ns/m}} $$ $$ \zeta \approx 0.4714 $$3. Calculate Damped Natural Frequency ($f_d$)
Calculate the damped angular frequency ($\omega_d$) using the previously found $\omega_n$ and $\zeta$:
$$ \omega_d = \omega_n \sqrt{1 - \zeta^2} $$ $$ \omega_d = 8.485 \, \text{rad/s} \times \sqrt{1 - (0.4714)^2} $$ $$ \omega_d = 8.485 \, \text{rad/s} \times \sqrt{1 - 0.2222} $$ $$ \omega_d = 8.485 \, \text{rad/s} \times \sqrt{0.7778} $$ $$ \omega_d = 8.485 \, \text{rad/s} \times 0.8819 $$ $$ \omega_d \approx 7.482 \, \text{rad/s} $$Finally, convert the damped angular frequency ($\omega_d$) to the damped natural frequency ($f_d$) in Hertz:
$$ f_d = \frac{\omega_d}{2\pi} = \frac{7.482 \, \text{rad/s}}{2\pi \, \text{rad}} $$ $$ f_d \approx \frac{7.482}{6.283} $$ $$ f_d \approx 1.191 \, \text{Hz} $$Final Answer Verification
The calculated values for the vehicle suspension system are:
- Damping Factor ($\zeta$): Approximately 0.471
- Damped Natural Frequency ($f_d$): Approximately 1.19 Hz
These results closely match the values presented in the first option.
Important Questions from Damping Coefficient and Damping Ratio
- 6ẍ + 9ẋ + 27x = 0 is the equation of motion for a damped vibration. The damping factor shall be:
- Ratio of actual to critical damping coefficient in forced vibrations is known as ________.
______ is defined as the ratio of the actual damping coefficient to a critical damping coefficient.
A spring-mass-damper system having single degree of freedom has a spring with strength 25 kN/m, mass 0.1 kg and coefficient of damping 40 N-s/m. The damping factor of the system will be
The damping ratio for a viscously damped spring mass system, governed by the relationship is \(m\frac{{{d^2}x}}{{d{t^2}}} + c\frac{{dx}}{{dt}} + kx = F\) given by
