Answered: Problem 5 The mass of the vehicle shown…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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$### Problem 5 The mass of the vehicle shown is 400 kg. This vehicle runs over a half-sine wave bump at 100 km/hr. Determine: (a) the response equation, (b) the maximum force exerted on the mass $m$ by the springs. Assume the damping ratio to be $\zeta = 0.4$. #### Diagram Explanation: The provided diagram consists of two parts: 1. **Bump Profile**: - The bump is described as a half sine wave. - The height of the bump is 10 cm. - The length of the bump along the road is 2 meters. 2. **Vehicle Suspension Model**: - The vehicle mass is denoted by $m$. - The vehicle rides over the bump at a speed of 100 km/hr. - The suspension system is modeled as a mass-spring-damper system. - There are two springs each with spring constant $k/2$, resulting in an equivalent spring constant $k$ in the vertical direction. - The damper in the system has a damping coefficient $c$. #### Equations: Given data: - Mass of the vehicle, $m = 400$ kg - Velocity of the vehicle, $v = 100$ km/hr = $\frac{100 \times 1000}{3600} \approx 27.78$ m/s - Damping ratio, $\zeta = 0.4$ We need to calculate: (a) Response equation (b) Maximum force exerted on the mass $m$ by the springs For more detailed solutions, refer to the principles of vibrational analysis and mechanical dynamics. This problem demonstrates the dynamics of a vehicle suspension system when interacting with road irregularities, which is crucial in vehicle design and comfort analysis.$

Transcribed Image Text:### Problem 5 The mass of the vehicle shown is 400 kg. This vehicle runs over a half-sine wave bump at 100 km/hr. Determine: (a) the response equation, (b) the maximum force exerted on the mass $m$ by the springs. Assume the damping ratio to be $\zeta = 0.4$. #### Diagram Explanation: The provided diagram consists of two parts: 1. **Bump Profile**: - The bump is described as a half sine wave. - The height of the bump is 10 cm. - The length of the bump along the road is 2 meters. 2. **Vehicle Suspension Model**: - The vehicle mass is denoted by $m$. - The vehicle rides over the bump at a speed of 100 km/hr. - The suspension system is modeled as a mass-spring-damper system. - There are two springs each with spring constant $k/2$, resulting in an equivalent spring constant $k$ in the vertical direction. - The damper in the system has a damping coefficient $c$. #### Equations: Given data: - Mass of the vehicle, $m = 400$ kg - Velocity of the vehicle, $v = 100$ km/hr = $\frac{100 \times 1000}{3600} \approx 27.78$ m/s - Damping ratio, $\zeta = 0.4$ We need to calculate: (a) Response equation (b) Maximum force exerted on the mass $m$ by the springs For more detailed solutions, refer to the principles of vibrational analysis and mechanical dynamics. This problem demonstrates the dynamics of a vehicle suspension system when interacting with road irregularities, which is crucial in vehicle design and comfort analysis.

### Equation and Solution for Forced and Free Vibration Analysis

The following is a detailed analysis of a system undergoing forced and free vibrations. The system’s equation of motion is given by:
\[ m\ddot{u} + c(\dot{u} - \dot{z}) + k(u - z) = 0 \]

Where:
- $ z = Z_0 \sin \Omega t $,
- $ \dot{z} = \Omega Z_0 \cos \Omega t $,
- $ \ddot{z} = -\Omega^2 Z_0 \sin \Omega t $.

This leads to the equation of motion as:
\[ m \ddot{u} + c \dot{u} = c\Omega Z_0 \cos \Omega t + k Z_0 \sin \Omega t \]

Or in terms of the simplified variables:
\[ \ddot{u} + 2 \zeta \omega_n \dot{u} + \omega_n^2 u = \frac{\Omega Z_0}{m} \cos \Omega t + \frac{k Z_0}{m} \sin \Omega t \]

Here:
- $ Z_0 = 0.1 m $
- $ T = \frac{l}{V} = \frac{2m}{27.778 m/s} = 0.072s $
- $ \Omega = \frac{2\pi}{T} = 87.27 \text{ rad/s} $

#### Free Vibration Analysis
1. **First Interval ( $ 0 < t < t_1 $ ):**
\[ u_{b_0}(t) = A \sin \Omega t + B \cos \Omega t \]
\[ u_{k_a}(t) = U e^{- \zeta \omega_n t} \cos(\omega_d t - \phi) \]
\[ u_{k_0}(t) = u_{b_0}(t) + \frac{U}{k} \]

With initial condition:
\[ u_{k_0}(0) = U_{k_0} \rightarrow u_{k_0}(0) = 0 \]

2. **Second Interval ( $ t > t_1 $ ):**
\[ u_{k_{max}}

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