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Industrial processes often require the damping of vibrations. Consider a workbench that uses four large ideal springs in place of traditional legs, with each spring supporting one corner of the workbench and exhibiting a force constant of k = 47500 N/m. This workbench is used when creating products that involve highly unstable chemicals. According to specifications, the acceleration of the manufacturing equipment must be held to less than 21.6 m/s² for the manufacturing process to proceed safely. The amplitudes of vibrations are expected to remain under 0.592 mm throughout. Given that the workbench and maufacturing equipment have a combined mass of m= 5.95 kg, find the maximum acceleration amax expected of the equipment and workbench. Ignore the mass of the springs. An engineer decides to damp the oscillations by applying one shock absorber to each spring. These shock absorbers apply a velocity-dependent drag force F = -bu that opposes the motion of the bench. For proper damping, the engineer decides that the motion of the bench should drop to half-amplitude within 0.479 s. In other words, within 0.479 s, the amplitude of motion should drop by a factor of one half. When purchasing shock absorbers, what damping coefficient b should the engineer specify to the industrial equipment supplier? Assume that the oscillations are underdamped.

Industrial processes often require the damping of vibrations. Consider a workbench that uses four large ideal springs in place of traditional legs, with each spring supporting one corner of the workbench and exhibiting a force constant of k = 47500 N/m. This workbench is used when creating products that involve highly unstable chemicals. According to specifications, the acceleration of the manufacturing equipment must be held to less than 21.6 m/s² for the manufacturing process to proceed safely. The amplitudes of vibrations are expected to remain under 0.592 mm throughout. Given that the workbench and maufacturing equipment have a combined mass of m= 5.95 kg, find the maximum acceleration amax expected of the equipment and workbench. Ignore the mass of the springs. An engineer decides to damp the oscillations by applying one shock absorber to each spring. These shock absorbers apply a velocity-dependent drag force F = -bu that opposes the motion of the bench. For proper damping, the engineer decides that the motion of the bench should drop to half-amplitude within 0.479 s. In other words, within 0.479 s, the amplitude of motion should drop by a factor of one half. When purchasing shock absorbers, what damping coefficient b should the engineer specify to the industrial equipment supplier? Assume that the oscillations are underdamped.

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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### Damping of Vibrations in Industrial Processes Industrial processes often require the damping of vibrations to ensure safety and accuracy. Consider a workbench that uses four large ideal springs instead of traditional legs, with each spring supporting one corner. These springs have a force constant of \( k = 47500 \, \text{N/m} \). This setup is used when handling products with highly unstable chemicals. #### Specifications: - **Maximum Safe Acceleration**: According to specifications, the acceleration of the manufacturing equipment must be less than \( 21.6 \, \text{m/s}^2 \) for safe operation. - **Amplitude of Vibrations**: Must remain under \( 0.592 \, \text{mm} \) at all times. - **Combined Mass**: The workbench and manufacturing equipment together have a mass of \( m = 5.95 \, \text{kg} \). - **Task**: Calculate the maximum acceleration \( a_{\text{max}} \) expected under these conditions, ignoring the mass of the springs. To further ensure stability, an engineer plans to install a shock absorber on each spring. These absorbers apply a force \( \vec{F} = -b\vec{v} \) that resists the motion of the bench. #### Shock Absorber Specifications: - **Requirement**: The amplitude of motion must decrease to half in \( 0.479 \, \text{s} \). - **Objective**: Identify the correct damping coefficient \( b \), assuming underdamped oscillations, for order specifications. **Problem to Solve**: Find the appropriate damping coefficient \( b \) (in \( \text{kg/s} \)) for these shock absorbers to ensure proper damping performance. --- This educational content is designed to help students and professionals understand the principles of damping and vibration control in industrial setups.
Transcribed Image Text:### Damping of Vibrations in Industrial Processes Industrial processes often require the damping of vibrations to ensure safety and accuracy. Consider a workbench that uses four large ideal springs instead of traditional legs, with each spring supporting one corner. These springs have a force constant of \( k = 47500 \, \text{N/m} \). This setup is used when handling products with highly unstable chemicals. #### Specifications: - **Maximum Safe Acceleration**: According to specifications, the acceleration of the manufacturing equipment must be less than \( 21.6 \, \text{m/s}^2 \) for safe operation. - **Amplitude of Vibrations**: Must remain under \( 0.592 \, \text{mm} \) at all times. - **Combined Mass**: The workbench and manufacturing equipment together have a mass of \( m = 5.95 \, \text{kg} \). - **Task**: Calculate the maximum acceleration \( a_{\text{max}} \) expected under these conditions, ignoring the mass of the springs. To further ensure stability, an engineer plans to install a shock absorber on each spring. These absorbers apply a force \( \vec{F} = -b\vec{v} \) that resists the motion of the bench. #### Shock Absorber Specifications: - **Requirement**: The amplitude of motion must decrease to half in \( 0.479 \, \text{s} \). - **Objective**: Identify the correct damping coefficient \( b \), assuming underdamped oscillations, for order specifications. **Problem to Solve**: Find the appropriate damping coefficient \( b \) (in \( \text{kg/s} \)) for these shock absorbers to ensure proper damping performance. --- This educational content is designed to help students and professionals understand the principles of damping and vibration control in industrial setups.
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