Act like a helpful tutor and exlain me :Chapter 1 — System Modeling
Problem 1.4 — The following figure shows a field-controlled motor. The motor inertia is I;
and the load inertia is I,. The motor a

Question

Act like a helpful tutor and exlain me :Chapter 1 — System Modeling
Problem 1.4 — The following figure shows a field-controlled motor. The motor inertia is I;
and the load inertia is I,. The motor and load are connected through a spring with rotational
stiffness (kr). We assume there is no translation motion between the motor and the load
but just rotational motion.
Task 1: Obtain a set of differential equations to model the entire system.
Hint #1: Similar to spring's linear dynamics (Force = Linear Stiffness x Compression, F =
kAx) , for spring's rotational dynamics, Torque = Rotational Stiffness x Rotational
Compression.
Hint #2: Find the equation of motion for motor and load using equation (Iw? = T) separately.
Hint #3: There should be 3 equations in total.
Task 2 (Optional): Describe the system using only a single differential equation
model with 8; as output and vy as input.

+

0, Ry

hr I

000
Motor Load
1 | oom so +]
kr c
T 0 0,

VIPSolutions · Accepted Answer

# Explanation of 1.4 – System Modeling

Let's break down the **system modeling** and guide you through setting up the differential equations step by step.

---

## **Understanding the System**

You have:
- **Field-controlled motor** (with inertia $ I_1 $)
- **Load** (with inertia $ I_2 $)
- **Rotational spring** (with stiffness $ k_T $) connecting motor and load
- Input: Field voltage ($ v_f $)
- Output: Load angular position ($ \theta_2 $)
- **No translational motion**—only rotational

### **Key Variables**
- $ \theta_1 $: Angular position of the motor
- $ \theta_2 $: Angular position of the load
- $ v_f $: Field voltage input to the motor

---

## **Task 1: Obtain Set of Differential Equations**

### **Step 1: Electrical Dynamics (Field Circuit)**

From the circuit:
$$
v_f = R_f i_f + L_f \frac{di_f}{dt}
$$
Where:
- $ R_f $: Field resistance
- $ L_f $: Field inductance
- $ i_f $: Field current

### **Step 2: Mechanical Dynamics (Motor and Load)**

#### **A. Motor (Inertia $ I_1 $)**

- **Equation of motion**: $ I_1 \ddot{\theta}_1 = T - T_s $
    - $ T $: Torque produced by the motor (proportional to field current, $ i_f $)
    - $ T_s $: Torque due to the spring
    
- **Spring torque** (like spring force): 
    $$
    T_s = k_T (\theta_1 - \theta_2)
    $$

- **Motor torque**: $ T = K_f i_f $, with $ K_f $ as the torque constant

So,
$$
I_1 \ddot{\theta}_1 = K_f i_f - k_T (\theta_1 - \theta_2)
$$

#### **B. Load (Inertia $ I_2 $)**

- **Equation of motion**: $ I_2 \ddot{\theta}_2 = T_s $ (assuming no damping or external load torque)

So,
$$
I_2 \ddot{\theta}_2 = k_T (\theta_1 - \theta_2)
$$

---

### **Summary of Differential Equations**

1. **Field Circuit:**
   $$
   v_f = R_f i_f + L_f \frac{di_f}{dt}
   $$
2. **Motor:**
   $$
   I_1 \ddot{\theta}_1 = K_f i_f - k_T (\theta_1 - \theta_2)
   $$
3. **Load:**
   $$
   I_2 \ddot{\theta}_2 = k_T (\theta_1 - \theta_2)
   $$

---

## **Task 2 (Optional): Single Differential Equation (with $ \theta_2 $ as Output, $ v_f $ as Input)**

### **Goal**: Express everything in terms of $ \theta_2 $ and $ v_f $.

**Outline of Steps:**
1. Express $ i_f $ from the field circuit equation.
2. Relate $ \theta_1 $ to $ \theta_2 $ using the load and motor equations.
3. Substitute back to eliminate intermediate variables.

#### **Step-by-step Outline**
1. **From load equation:**  
   $ I_2 \ddot{\theta}_2 = k_T (\theta_1 - \theta_2) $  
   $ \Rightarrow \theta_1 = \theta_2 + \frac{I_2}{k_T} \ddot{\theta}_2 $
2. **Substitute $ \theta_1 $ into motor equation:**  
   $ I_1 \ddot{\theta}_1 = K_f i_f - k_T (\theta_1 - \theta_2) $  
   But $ (\theta_1 - \theta_2) = \frac{I_2}{k_T} \ddot{\theta}_2 $
3. **Combine and substitute $ i_f $ from field circuit equation.**

This leads to a **third-order** single differential equation relating $ \theta_2 $ and $ v_f $.

---

## **Summary Table**

| Equation Type     | Equation                                                                 |
|-------------------|--------------------------------------------------------------------------|
| Field Circuit     | $ v_f = R_f i_f + L_f \frac{di_f}{dt} $                               |
| Motor             | $ I_1 \ddot{\theta}_1 = K_f i_f - k_T (\theta_1 - \theta_2) $         |
| Load              | $ I_2 \ddot{\theta}_2 = k_T (\theta_1 - \theta_2) $                   |

---

### **Diagram Reference**
- $ v_f $ is the input voltage, $ \theta_2 $ is the output (load position).

---

## **Summary**

- **Task 1**: Three equations—one for the field circuit, one for the motor, and one for the load.
- **Task 2**: Combine the equations to relate $ v_f $ (input) to $ \theta_2 $ (output) by eliminating intermediate variables.

Let me know if you want to see the single combined equation for Task 2 worked out explicitly!

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