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Seen in Figure 1, a small cart (with mass m and position x2) is riding inside a larger cart, which is massless, and has a position ₁. A spring and a damper in parallel connect the larger cart to the smaller cart, and another spring connects the exterior of the large massless cart to an input position, £,. For similar to the one in Homework 2, Problem 1. It has the same equations of

Seen in Figure 1, a small cart (with mass m and position x2) is riding inside a larger cart, which is massless, and has a position ₁. A spring and a damper in parallel connect the larger cart to the smaller cart, and another spring connects the exterior of the large massless cart to an input position, £,. For similar to the one in Homework 2, Problem 1. It has the same equations of

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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### Educational Material: Mechanical System Analysis **Figure Explanation:** The image depicts a mechanical system comprising two carts, where: - A small cart with mass \( m \) and position \( x_2 \) is situated inside a larger, massless cart which has a position \( x_1 \). - The larger cart is connected to the smaller cart using a spring (\( k \)) and a damper (\( c \)) in parallel. - Another spring connects the exterior of the large cart to an input position, \( x_u \). **Diagram Description:** - The small cart is represented with \( m \) and has wheels on a horizontal surface. - The outer cart includes springs and a damper connected in parallel, illustrated in the diagram. - Directions \( x_1 \) and \( x_2 \) are indicated with arrows to represent the motion of each cart. **Tasks:** A. **Draw the Free Body Diagram (FBD) and Write Equations of Motion:** - Analyze the forces acting on each cart, including spring forces, damping forces, and external inputs. - Establish the equations of motion based on Newton's laws for each mass. B. **State-Space Representation:** - Derive the state-space equations using \( x_1 \) and \( x_2 \) and their derivatives. - Translate the mechanical system into a second-order matrix form for easier analysis. This educational material guides students through modeling complex mechanical systems, emphasizing the use of free body diagrams and state-space analysis to comprehend dynamic behaviors.
Transcribed Image Text:### Educational Material: Mechanical System Analysis **Figure Explanation:** The image depicts a mechanical system comprising two carts, where: - A small cart with mass \( m \) and position \( x_2 \) is situated inside a larger, massless cart which has a position \( x_1 \). - The larger cart is connected to the smaller cart using a spring (\( k \)) and a damper (\( c \)) in parallel. - Another spring connects the exterior of the large cart to an input position, \( x_u \). **Diagram Description:** - The small cart is represented with \( m \) and has wheels on a horizontal surface. - The outer cart includes springs and a damper connected in parallel, illustrated in the diagram. - Directions \( x_1 \) and \( x_2 \) are indicated with arrows to represent the motion of each cart. **Tasks:** A. **Draw the Free Body Diagram (FBD) and Write Equations of Motion:** - Analyze the forces acting on each cart, including spring forces, damping forces, and external inputs. - Establish the equations of motion based on Newton's laws for each mass. B. **State-Space Representation:** - Derive the state-space equations using \( x_1 \) and \( x_2 \) and their derivatives. - Translate the mechanical system into a second-order matrix form for easier analysis. This educational material guides students through modeling complex mechanical systems, emphasizing the use of free body diagrams and state-space analysis to comprehend dynamic behaviors.
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