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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
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
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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
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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