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stess, and Seen in Figure 1, a small cart (with mass m and position 2) is riding inside a 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 this problem, the diagram is similar to the one in Homework 2, Problem 1. It has the same equations of motion. You are not required to re-derive the equations of motion - please reference the solutions. Given this, ↑ ww + C m X2 Given equation of motion: ⒸSmall cort: x₂ + ((x₂-x₂)+(x₂-x)=0 Ⓒlarge carr: K√(x₂-2x1 + x ₁ ) + C (x₂-x) =

stess, and Seen in Figure 1, a small cart (with mass m and position 2) is riding inside a 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 this problem, the diagram is similar to the one in Homework 2, Problem 1. It has the same equations of motion. You are not required to re-derive the equations of motion - please reference the solutions. Given this, ↑ ww + C m X2 Given equation of motion: ⒸSmall cort: x₂ + ((x₂-x₂)+(x₂-x)=0 Ⓒlarge carr: K√(x₂-2x1 + x ₁ ) + C (x₂-x) =

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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**Mechanical System Modeling and Equations of Motion** In the figure presented, a small cart with a mass \( m \) and position \( x_2 \) rides inside a larger massless cart with a position \( x_1 \). A spring and a damper, arranged in parallel, link the larger cart to the smaller cart. Additionally, a spring connects the exterior of the larger cart to an input position, \( x_u \). ### Diagram Explanation: **Figure 1: Schematic of the mechanical system to model in Problem 1** The diagram illustrates the configuration of springs (with stiffness \( k \)) and a damper (with damping coefficient \( c \)), highlighting the interaction between the carts and their components. ### Given Equations of Motion: 1. **For the Small Cart:** \[ m \ddot{x}_2 + c(\dot{x}_2 - \dot{x}_1) + k(x_2 - x_1) = 0 \] 2. **For the Large Cart:** \[ k(x_1 - x_2 + x_u) + c(\dot{x}_1 - \dot{x}_2) = 0 \] ### Tasks: A. Determine the state-space equations for the system in matrix form, utilizing \( x_1 \) and \( x_2 \) along with their derivatives. B. Represent the mechanical system in second-order matrix form using \( x_1 \) and \( x_2 \) and their derivatives.
Transcribed Image Text:**Mechanical System Modeling and Equations of Motion** In the figure presented, a small cart with a mass \( m \) and position \( x_2 \) rides inside a larger massless cart with a position \( x_1 \). A spring and a damper, arranged in parallel, link the larger cart to the smaller cart. Additionally, a spring connects the exterior of the larger cart to an input position, \( x_u \). ### Diagram Explanation: **Figure 1: Schematic of the mechanical system to model in Problem 1** The diagram illustrates the configuration of springs (with stiffness \( k \)) and a damper (with damping coefficient \( c \)), highlighting the interaction between the carts and their components. ### Given Equations of Motion: 1. **For the Small Cart:** \[ m \ddot{x}_2 + c(\dot{x}_2 - \dot{x}_1) + k(x_2 - x_1) = 0 \] 2. **For the Large Cart:** \[ k(x_1 - x_2 + x_u) + c(\dot{x}_1 - \dot{x}_2) = 0 \] ### Tasks: A. Determine the state-space equations for the system in matrix form, utilizing \( x_1 \) and \( x_2 \) along with their derivatives. B. Represent the mechanical system in second-order matrix form using \( x_1 \) and \( x_2 \) and their derivatives.
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