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Using graphical and analytical methods, determine the torque, T12, required to drive slider 6 of Figure against a load of P = 100 lb at a crank angle of 0 = 30°. RA0₂ = 62.5 mm, R0₂04 = 400 mm, RBc = 200 mm, RBO4 = 150 mm BC

Using graphical and analytical methods, determine the torque, T12, required to drive slider 6 of Figure against a load of P = 100 lb at a crank angle of 0 = 30°. RA0₂ = 62.5 mm, R0₂04 = 400 mm, RBc = 200 mm, RBO4 = 150 mm BC

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
Problem 1.1MA
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**Title:** Determining the Required Torque for Crank Mechanism Operation **Objective:** To determine the torque \( T_{12} \) necessary to drive slider 6 against a load of \( P = 100 \, \text{lb} \) at a crank angle of \( \theta = 30^\circ \) using graphical and analytical methods. **Given Parameters:** - \( \vec{R}_{A_{O_2}} = 62.5 \, \text{mm} \) - \( \vec{R}_{O_2O_4} = 400 \, \text{mm} \) - \( \vec{R}_{BC} = 200 \, \text{mm} \) - \( \vec{R}_{B_{O_4}} = 150 \, \text{mm} \) **Diagram Explanation:** 1. **Mechanism Components:** - A crank (labeled as line 2) connected to a rotating joint at \( O_2 \). - A linkage (labeled as line 3) connected to the crank, transferring motion to a slider (labeled as 6) through a link (labeled as line 4). 2. **Positions and Distances:** - Slider 6 is horizontally displaced to the left by 250 mm from the center of rotation \( O_2 \). - A vertical linkage from \( B \) to \( O_4 \) is 150 mm below \( O_2 \). 3. **Angles:** - \( \theta_2 \) is marked as the angle of the crank which is given as 30°. 4. **Torque Indication:** - The direction of torque \( T_{12} \) is counterclockwise on the crank around joint \( O_2 \). **Application:** This system exemplifies analyzing a simple slider-crank mechanism often found in internal combustion engines. By determining the precise torque required at a specific crank angle, engineers can ensure the mechanism operates effectively under the specified load conditions.
Transcribed Image Text:**Title:** Determining the Required Torque for Crank Mechanism Operation **Objective:** To determine the torque \( T_{12} \) necessary to drive slider 6 against a load of \( P = 100 \, \text{lb} \) at a crank angle of \( \theta = 30^\circ \) using graphical and analytical methods. **Given Parameters:** - \( \vec{R}_{A_{O_2}} = 62.5 \, \text{mm} \) - \( \vec{R}_{O_2O_4} = 400 \, \text{mm} \) - \( \vec{R}_{BC} = 200 \, \text{mm} \) - \( \vec{R}_{B_{O_4}} = 150 \, \text{mm} \) **Diagram Explanation:** 1. **Mechanism Components:** - A crank (labeled as line 2) connected to a rotating joint at \( O_2 \). - A linkage (labeled as line 3) connected to the crank, transferring motion to a slider (labeled as 6) through a link (labeled as line 4). 2. **Positions and Distances:** - Slider 6 is horizontally displaced to the left by 250 mm from the center of rotation \( O_2 \). - A vertical linkage from \( B \) to \( O_4 \) is 150 mm below \( O_2 \). 3. **Angles:** - \( \theta_2 \) is marked as the angle of the crank which is given as 30°. 4. **Torque Indication:** - The direction of torque \( T_{12} \) is counterclockwise on the crank around joint \( O_2 \). **Application:** This system exemplifies analyzing a simple slider-crank mechanism often found in internal combustion engines. By determining the precise torque required at a specific crank angle, engineers can ensure the mechanism operates effectively under the specified load conditions.
**Title: Analysis of Forces in a Four-Bar Linkage System** **Introduction:** This resource provides methods to determine the forces acting on the ground and the required torque, \( T_{12} \), to maintain static equilibrium in a four-bar linkage system. Both graphical and analytical methods are used. **Image Description:** The diagram illustrates a four-bar linkage mechanism comprising links labeled as \( AB \), \( BO_4 \), \( CO_4 \), \( DO_4 \), and \( O_2O_4 \). Specific measurements are provided for the vectors representing each link: - \( \vec{R}_{A O_2} = 87.5 \, mm \) - \( \vec{R}_{BA} = \vec{R}_{BO_4} = 150 \, mm \) - \( \vec{R}_{CO_4} = 100 \, mm \) - \( \vec{R}_{DO_4} = 50 \, mm \) - \( \vec{R}_{O_2O_4} = 175 \, mm \) **Diagram Details:** - **Points and Angles:** The points involved are \( A \), \( B \), \( O_2 \), \( O_4 \), and \( D \). - **Linkage Layout:** The linkage chain forms a closed loop comprising bars \( AB \), \( BO_4 \), \( O_4O_2 \), and \( AO_2 \). - **Forces and Torques:** An external force \( P \) acts at point \( D \), and the torque \( T_{12} \) is applied around \( O_2 \) to maintain equilibrium. The specified angles between links, \( 105.1^\circ \) and \( 152.4^\circ \), are crucial in determining force directions. **Conclusion:** By using the given dimensions and angles, the force and torque can be computed for static equilibrium. This foundational understanding applies particularly in mechanical engineering and robotics, where precise movement and stability are crucial. **Note:** For further understanding, students and practitioners can recreate the diagram for hands-on practice and symbolic computations.
Transcribed Image Text:**Title: Analysis of Forces in a Four-Bar Linkage System** **Introduction:** This resource provides methods to determine the forces acting on the ground and the required torque, \( T_{12} \), to maintain static equilibrium in a four-bar linkage system. Both graphical and analytical methods are used. **Image Description:** The diagram illustrates a four-bar linkage mechanism comprising links labeled as \( AB \), \( BO_4 \), \( CO_4 \), \( DO_4 \), and \( O_2O_4 \). Specific measurements are provided for the vectors representing each link: - \( \vec{R}_{A O_2} = 87.5 \, mm \) - \( \vec{R}_{BA} = \vec{R}_{BO_4} = 150 \, mm \) - \( \vec{R}_{CO_4} = 100 \, mm \) - \( \vec{R}_{DO_4} = 50 \, mm \) - \( \vec{R}_{O_2O_4} = 175 \, mm \) **Diagram Details:** - **Points and Angles:** The points involved are \( A \), \( B \), \( O_2 \), \( O_4 \), and \( D \). - **Linkage Layout:** The linkage chain forms a closed loop comprising bars \( AB \), \( BO_4 \), \( O_4O_2 \), and \( AO_2 \). - **Forces and Torques:** An external force \( P \) acts at point \( D \), and the torque \( T_{12} \) is applied around \( O_2 \) to maintain equilibrium. The specified angles between links, \( 105.1^\circ \) and \( 152.4^\circ \), are crucial in determining force directions. **Conclusion:** By using the given dimensions and angles, the force and torque can be computed for static equilibrium. This foundational understanding applies particularly in mechanical engineering and robotics, where precise movement and stability are crucial. **Note:** For further understanding, students and practitioners can recreate the diagram for hands-on practice and symbolic computations.
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