Find the variation of the displacement with time, x(t), of a damped single- degree-of-freedom system with m=10 kg, k=10,000 N/m, and 3=0.1 for the following initial conditions:
Find the variation of the displacement with time, x(t), of a damped single- degree-of-freedom system with m=10 kg, k=10,000 N/m, and 3=0.1 for the following initial conditions:
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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ANSWER problem 2.121.Use x(0)=.1m and x(dot)(0)=2 as a template for answering the problem stated problem 2.121. And note theta tells you which quadrant the tan inverse value will fall in so for example theta with two positive values so the angle inputted will be in quadrant 1 (in the unit circle).Negative value from tan inverse means subtract the inverse tan value from larger value and positive value means add to the smaller value. EX: quad 2 (in the unit circle) smaller value is 90 and larger value is 180
Transcribed Image Text:**Problem 2.121: Variation of Displacement**
Find the **variation of the displacement** with time, \( x(t) \), of a damped single-degree-of-freedom system with \( m = 10 \, \text{kg} \), \( k = 10{,}000 \, \text{N/m} \), and \( \zeta = 0.1 \) for the following initial conditions:
(a) \( x(0) = 0.2 \, \text{m}, \hspace{0.2cm} \dot{x}(0) = 0 \)
(b) \( x(0) = -0.2 \, \text{m}, \hspace{0.2cm} \dot{x}(0) = 0 \)
(c) \( x(0) = 0, \hspace{0.2cm} \dot{x}(0) = 0.2 \, \left( \frac{\text{m}}{\text{sec}} \right) \)
![### Single-Degree-of-Freedom System Analysis
**Parameters and Constants:**
- **Mass, \( m \):** 1 kg
- **Damping coefficient, \( c \):** 5 N-sec/m
- **Stiffness, \( k \):** 16 N/m
For the system described, the response is analyzed for the following initial conditions:
#### Initial Conditions:
(a) \( x(0) = 0.1 \, m \), \( \dot{x}(0) = 2 \, \frac{m}{sec} \)
(b) \( x(0) = -0.1 \, m \), \( \dot{x}(0) = -2 \, \frac{m}{sec} \)
**Differential Equation of Motion:**
\[ m\ddot{x} + c\dot{x} + kx = 0 \]
\[ \to \ddot{x} + 5\dot{x} + 16x = 0 \]
\[ \to x = \frac{-5 \pm \sqrt{25 - 4(16)}}{2} = \frac{-5 \pm i\sqrt{39}}{2} \]
#### System Response:
\[ x(t) = X_0 e^{-\zeta\omega_nt} \sin(\omega_d t + \phi_0) \]
Where:
\[ X_0 = \frac{\sqrt{x_0^2 \omega_n^2 + \dot{x_0}^2 + 2\zeta\omega_nx_0\dot{x_0}}}{\omega_d} \]
\[ \phi_0 = \tan^{-1} \left( \frac{x_0 \omega_d}{\dot{x_0} + \zeta\omega_nx_0} \right) \]
### Parameters Calculation:
- **Natural Frequency:**
\[ \omega_n = \sqrt{\frac{k}{m}} = \sqrt{\frac{16}{1}} = 4 \, \frac{rad}{sec} \]
- **Critical Damping Coefficient:**
\[ c_c = 2\sqrt{km} = 2\sqrt{16(1)} = 8 \, \frac{N \cdot sec}{m} \]
- **Damping Ratio:**
\[ \zeta](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20000a63-5a8a-4b5e-adbd-85ecbf6c298b%2Fea7486da-9580-4184-beb9-e5ba48288b23%2Fgg3dpmb_processed.png&w=3840&q=75)
Transcribed Image Text:### Single-Degree-of-Freedom System Analysis
**Parameters and Constants:**
- **Mass, \( m \):** 1 kg
- **Damping coefficient, \( c \):** 5 N-sec/m
- **Stiffness, \( k \):** 16 N/m
For the system described, the response is analyzed for the following initial conditions:
#### Initial Conditions:
(a) \( x(0) = 0.1 \, m \), \( \dot{x}(0) = 2 \, \frac{m}{sec} \)
(b) \( x(0) = -0.1 \, m \), \( \dot{x}(0) = -2 \, \frac{m}{sec} \)
**Differential Equation of Motion:**
\[ m\ddot{x} + c\dot{x} + kx = 0 \]
\[ \to \ddot{x} + 5\dot{x} + 16x = 0 \]
\[ \to x = \frac{-5 \pm \sqrt{25 - 4(16)}}{2} = \frac{-5 \pm i\sqrt{39}}{2} \]
#### System Response:
\[ x(t) = X_0 e^{-\zeta\omega_nt} \sin(\omega_d t + \phi_0) \]
Where:
\[ X_0 = \frac{\sqrt{x_0^2 \omega_n^2 + \dot{x_0}^2 + 2\zeta\omega_nx_0\dot{x_0}}}{\omega_d} \]
\[ \phi_0 = \tan^{-1} \left( \frac{x_0 \omega_d}{\dot{x_0} + \zeta\omega_nx_0} \right) \]
### Parameters Calculation:
- **Natural Frequency:**
\[ \omega_n = \sqrt{\frac{k}{m}} = \sqrt{\frac{16}{1}} = 4 \, \frac{rad}{sec} \]
- **Critical Damping Coefficient:**
\[ c_c = 2\sqrt{km} = 2\sqrt{16(1)} = 8 \, \frac{N \cdot sec}{m} \]
- **Damping Ratio:**
\[ \zeta
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