(2) A mechanical system is modeled by the system of ODE's. For this system choose x= X₁ , consider the output to be y, and do the following: Determine the state-space matrices A, B, and C Determine the characteristic equation mÿ+k₁y+k₂(y−z)=F c₂ż-k₂(y-z)=0
(2) A mechanical system is modeled by the system of ODE's. For this system choose x= X₁ , consider the output to be y, and do the following: Determine the state-space matrices A, B, and C Determine the characteristic equation mÿ+k₁y+k₂(y−z)=F c₂ż-k₂(y-z)=0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 2: Mechanical System Modeled by ODEs**
A mechanical system is modeled by the system of Ordinary Differential Equations (ODEs). For this system, choose
\[
\vec{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} y \\ \dot{y} \\ z \end{bmatrix}
\]
Consider the output to be \( y \), and perform the following tasks:
- **Determine the state-space matrices \( A, B, \) and \( C \)**
- **Determine the characteristic equation**
The system is described by the following equations:
\[ m\ddot{y} + k_1y + k_2(y - z) = F \]
\[ c_2\dot{z} - k_2(y - z) = 0 \]
**Explanation:**
The problem involves deriving state-space representations and finding the characteristic equation of a mechanical system modeled by differential equations. The system involves two equations that relate the variables \( y \), \( \dot{y} \), \( z \), and their derivatives. The main goal is to convert these into a form where state-space matrices \( A, B, \) and \( C \) can be identified. These matrices help in analyzing the dynamic behavior of the system.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdfe80350-6fa1-469d-aae2-95b4dd3aa499%2Fe9b72095-a727-4014-829b-fd4b4104f23b%2Fvvs5y4e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 2: Mechanical System Modeled by ODEs**
A mechanical system is modeled by the system of Ordinary Differential Equations (ODEs). For this system, choose
\[
\vec{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} y \\ \dot{y} \\ z \end{bmatrix}
\]
Consider the output to be \( y \), and perform the following tasks:
- **Determine the state-space matrices \( A, B, \) and \( C \)**
- **Determine the characteristic equation**
The system is described by the following equations:
\[ m\ddot{y} + k_1y + k_2(y - z) = F \]
\[ c_2\dot{z} - k_2(y - z) = 0 \]
**Explanation:**
The problem involves deriving state-space representations and finding the characteristic equation of a mechanical system modeled by differential equations. The system involves two equations that relate the variables \( y \), \( \dot{y} \), \( z \), and their derivatives. The main goal is to convert these into a form where state-space matrices \( A, B, \) and \( C \) can be identified. These matrices help in analyzing the dynamic behavior of the system.
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