4- Represent the system shown in Figure P4 in state space where the output is OL(t). T(t) N₁ = 30 2 N-m/rad 3 N-m-s/rad oooo N-m/rad N3 = 10 farm FIGURE P4 10 N₂= 300 I▬▬▬▬▬▬▬▬ N4 = 100 200 N-m-s/rad
4- Represent the system shown in Figure P4 in state space where the output is OL(t). T(t) N₁ = 30 2 N-m/rad 3 N-m-s/rad oooo N-m/rad N3 = 10 farm FIGURE P4 10 N₂= 300 I▬▬▬▬▬▬▬▬ N4 = 100 200 N-m-s/rad
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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![### Understanding Mechanical System Dynamics
#### Problem Statement:
4. Represent the system shown in **Figure P4** in state space where the output is \(\theta_L(t)\).
---
#### Description of Figure P4:
Figure P4 depicts a mechanical system with interconnected components that include springs, dampers, and mass elements. The system details are as follows:
1. **Input Torque**:
- \( T(t) \) is the applied torque input to the system.
2. **Spring Constants**:
- Two springs are shown with the following spring constants:
- Spring 1: \( \frac{1}{10} \, \text{N-m/rad} \)
- Spring 2: \( 2 \, \text{N-m/rad} \)
3. **Damper Constants**:
- Three dampers are included in the system with the following constants:
- Damper 1: \( 3 \, \text{N-m-s/rad} \)
- Damper 2: \( 200 \, \text{N-m-s/rad} \)
4. **Inertial Elements (Moments of Inertia)**:
- The system consists of four inertia elements:
- \( N_1 = 30 \)
- \( N_2 = 300 \)
- \( N_3 = 10 \)
- \( N_4 = 100 \)
5. **Output Angular Displacement**:
- The output of interest is the angular displacement \(\theta_L(t)\).
---
#### Explanation:
The mechanical diagram can be represented as a combination of interconnected components with specific properties that define their relationship in terms of forces and displacements. Within the system:
- Springs represent elements that store elastic energy and exhibit a linear relationship between torque and angular displacement.
- Dampers represent elements that exhibit a linear relationship between torque and angular velocity, thus dissipating energy.
- Inertia elements represent the resistance to changes in rotational motion.
The goal is to translate this mechanical system into a state-space representation, which necessitates defining equations of motion for each component and combining them into a matrix form that relates input \(T(t)\) to the output \(\theta_L(t)\).
---
To solve this problem, we need to derive the differential equations for the system, identify the state variables, and represent these equations in matrix form for the state-space](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9dac685f-9be9-4df9-bb3b-1195be9f12e4%2Fb143b066-fd7e-4971-9fc9-a558b538b2b2%2Fjfisvcq_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Mechanical System Dynamics
#### Problem Statement:
4. Represent the system shown in **Figure P4** in state space where the output is \(\theta_L(t)\).
---
#### Description of Figure P4:
Figure P4 depicts a mechanical system with interconnected components that include springs, dampers, and mass elements. The system details are as follows:
1. **Input Torque**:
- \( T(t) \) is the applied torque input to the system.
2. **Spring Constants**:
- Two springs are shown with the following spring constants:
- Spring 1: \( \frac{1}{10} \, \text{N-m/rad} \)
- Spring 2: \( 2 \, \text{N-m/rad} \)
3. **Damper Constants**:
- Three dampers are included in the system with the following constants:
- Damper 1: \( 3 \, \text{N-m-s/rad} \)
- Damper 2: \( 200 \, \text{N-m-s/rad} \)
4. **Inertial Elements (Moments of Inertia)**:
- The system consists of four inertia elements:
- \( N_1 = 30 \)
- \( N_2 = 300 \)
- \( N_3 = 10 \)
- \( N_4 = 100 \)
5. **Output Angular Displacement**:
- The output of interest is the angular displacement \(\theta_L(t)\).
---
#### Explanation:
The mechanical diagram can be represented as a combination of interconnected components with specific properties that define their relationship in terms of forces and displacements. Within the system:
- Springs represent elements that store elastic energy and exhibit a linear relationship between torque and angular displacement.
- Dampers represent elements that exhibit a linear relationship between torque and angular velocity, thus dissipating energy.
- Inertia elements represent the resistance to changes in rotational motion.
The goal is to translate this mechanical system into a state-space representation, which necessitates defining equations of motion for each component and combining them into a matrix form that relates input \(T(t)\) to the output \(\theta_L(t)\).
---
To solve this problem, we need to derive the differential equations for the system, identify the state variables, and represent these equations in matrix form for the state-space
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