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The amplitude of vibration for the system shown is observed to decrease to 25% of its initial value after five cycles. If m= 10th and k=20% in. Find C # Ljc k Схл M => M Rx T=bmx ² v=kx² 00 F(X)= mxex-kx (>= mx + kx- HIL Meq = M + ²/2

The amplitude of vibration for the system shown is observed to decrease to 25% of its initial value after five cycles. If m= 10th and k=20% in. Find C # Ljc k Схл M => M Rx T=bmx ² v=kx² 00 F(X)= mxex-kx (>= mx + kx- HIL Meq = M + ²/2

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
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The amplitude of vibration for the system is observed to decrease by 25% of its initial value after 5 cycles. If m=10^# and k=20#/in. Find c Everything under the initial drawing is my scratch work, please start with that if possible.
### Vibration Analysis of a Mechanical System #### Problem Description The amplitude of vibration for the system shown is observed to decrease to 25% of its initial value after five cycles. Given \( m = 10 \, \text{lb} \) and \( k = 20 \, \text{lb/in.} \), find \( C \). #### Diagram Explanation - The system consists of a mass \( m \) connected to both a spring \( k \) and a dashpot \( C \). - A force \( F(t) \) is applied to the mass, resulting in motion described by the system’s differential equations. #### Equations 1. **Equation of Motion:** \[ F(t) = m \ddot{x} + c \dot{x} + kx \] - Here, \( \ddot{x} \) and \( \dot{x} \) are the acceleration and velocity of the mass, respectively. 2. **Simplified Formulation:** \[ C > 0 = m \ddot{x} + kx = 0 \] 3. **Mechanical Energy Components:** - Kinetic Energy \( T = \frac{1}{2}m\dot{x}^2 \) - Potential Energy \( V = \frac{1}{2}kx^2 \) 4. **Effective Mass:** \[ m_{\text{eq}} = m + \frac{k}{g} \] #### Explanation The system undergoes damped harmonic motion due to the presence of the dashpot \( C \). The objective is to determine the damping coefficient \( C \) that causes the amplitude to reduce to 25% of its original in five cycles. This involves solving the differential motion equation and applying the conditions for damped vibrations. This analysis is crucial for understanding how mechanical systems behave under oscillating forces and how damping influences vibrational amplitude and system stability.
Transcribed Image Text:### Vibration Analysis of a Mechanical System #### Problem Description The amplitude of vibration for the system shown is observed to decrease to 25% of its initial value after five cycles. Given \( m = 10 \, \text{lb} \) and \( k = 20 \, \text{lb/in.} \), find \( C \). #### Diagram Explanation - The system consists of a mass \( m \) connected to both a spring \( k \) and a dashpot \( C \). - A force \( F(t) \) is applied to the mass, resulting in motion described by the system’s differential equations. #### Equations 1. **Equation of Motion:** \[ F(t) = m \ddot{x} + c \dot{x} + kx \] - Here, \( \ddot{x} \) and \( \dot{x} \) are the acceleration and velocity of the mass, respectively. 2. **Simplified Formulation:** \[ C > 0 = m \ddot{x} + kx = 0 \] 3. **Mechanical Energy Components:** - Kinetic Energy \( T = \frac{1}{2}m\dot{x}^2 \) - Potential Energy \( V = \frac{1}{2}kx^2 \) 4. **Effective Mass:** \[ m_{\text{eq}} = m + \frac{k}{g} \] #### Explanation The system undergoes damped harmonic motion due to the presence of the dashpot \( C \). The objective is to determine the damping coefficient \( C \) that causes the amplitude to reduce to 25% of its original in five cycles. This involves solving the differential motion equation and applying the conditions for damped vibrations. This analysis is crucial for understanding how mechanical systems behave under oscillating forces and how damping influences vibrational amplitude and system stability.
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