Introductory Circuit Analysis (13th Edition)
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ISBN: 9780133923605
Author: Robert L. Boylestad
Publisher: PEARSON
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![**Educational Content: Mechanical and Electrical System Analysis**
**Mechanical System Description:**
The mechanical system consists of two masses, \( M_1 \) and \( M_2 \), each weighing 1 kg. They are connected in series with springs \( k_1 \) and \( k_2 \) both having spring constants of 1 N/m. The system is subjected to damping forces \( fv_1 = 2 \, \text{N·s/m} \), \( fv_2 = 1 \, \text{N·s/m} \), and \( fv_3 = 1 \, \text{N·s/m} \). The velocities of the masses are \( v_1(t) \) and \( v_2(t) \). An external force \( f(t) \) is applied to the system.
**a) Differential Equations:**
1. \( f_{v1} = \frac{dv_1}{dt} (k_1, M_1) \)
2. \( f_{v2} = \frac{dv_2}{dt} (k_2, k_1, M_1, M_2) \)
3. \( f_{v3} = \frac{dv_3}{dt} \left( \frac{k_2}{M_2} \right) \)
The expression for the applied force \( f(t) \) is derived as:
\[ f(t) = \frac{dv_1}{dt} - k_1 M_1 + \frac{dv_2}{dt} (k_2, k_1, M_1, M_2) + \frac{dv_3}{dt} \left( \frac{k_1}{M_2} \right) \]
**b) Equivalent Electrical Circuit:**
The equivalent electrical circuit is drawn with the following components:
- An inductor \( M_1 \) representing mass \( M_1 \).
- Two capacitors representing the spring constants and damping.
- A voltage source, \( f(t) = V_{\text{out}} \), replaces the external force.
Note that the component remarked as "A gate, Not a Capacitor" replaces an element in the system.
This representation helps in understanding the analogous relationship between mechanical damped spring-mass systems and RLC circuits in electrical engineering.](https://content.bartleby.com/qna-images/question/ecd9d238-92a1-4a47-af9e-c758e887edd1/c68003df-7cfc-426d-bfa4-3cdb3f86f263/mlhfwjc_processed.png)
Transcribed Image Text:**Educational Content: Mechanical and Electrical System Analysis**
**Mechanical System Description:**
The mechanical system consists of two masses, \( M_1 \) and \( M_2 \), each weighing 1 kg. They are connected in series with springs \( k_1 \) and \( k_2 \) both having spring constants of 1 N/m. The system is subjected to damping forces \( fv_1 = 2 \, \text{N·s/m} \), \( fv_2 = 1 \, \text{N·s/m} \), and \( fv_3 = 1 \, \text{N·s/m} \). The velocities of the masses are \( v_1(t) \) and \( v_2(t) \). An external force \( f(t) \) is applied to the system.
**a) Differential Equations:**
1. \( f_{v1} = \frac{dv_1}{dt} (k_1, M_1) \)
2. \( f_{v2} = \frac{dv_2}{dt} (k_2, k_1, M_1, M_2) \)
3. \( f_{v3} = \frac{dv_3}{dt} \left( \frac{k_2}{M_2} \right) \)
The expression for the applied force \( f(t) \) is derived as:
\[ f(t) = \frac{dv_1}{dt} - k_1 M_1 + \frac{dv_2}{dt} (k_2, k_1, M_1, M_2) + \frac{dv_3}{dt} \left( \frac{k_1}{M_2} \right) \]
**b) Equivalent Electrical Circuit:**
The equivalent electrical circuit is drawn with the following components:
- An inductor \( M_1 \) representing mass \( M_1 \).
- Two capacitors representing the spring constants and damping.
- A voltage source, \( f(t) = V_{\text{out}} \), replaces the external force.
Note that the component remarked as "A gate, Not a Capacitor" replaces an element in the system.
This representation helps in understanding the analogous relationship between mechanical damped spring-mass systems and RLC circuits in electrical engineering.
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