23000064940 *} } [ < € ] {* } + [ ^** + m₂ + m₂ m₂ m₂ m₁ =1kg m₂ = 2kg k₁ = k₁= k₂ = 1¾ 2₁ Determine (analytically do not use Matlab): 1) The characteristic equation 2) The natural frequencies 3) The mode shapes normalized to the largest value = 1.0 4) Sketch the mode shapes 6) Determine the solutions [10] X₂0 Wis using the modal method. 7) Plot your results using MatLab. 5) Show the mode shapes are orthogonal with respect to the mass matrix x₂ (1) and x₂ (1) for -.7207 rad/s ₁₂ Wn₂ = .1.7552 rad/s 1.0 -0.1754) 11 = 1 [(k₂ + 4k₂ + k₂) −4k₂ 11 -4k₂ 2 = nm 30-4 [0.701 1.0 x₁ (t) = 7.1664 sin(.7207t-.00719)+8.7643 sin 1.1552t+2.9678) x₂ (t)=10.2149 sin(.7207t-.00719)-1.5372 sin 1.1552t +2.9678) {*}={100mm/sec M(t) M(t) R
23000064940 *} } [ < € ] {* } + [ ^** + m₂ + m₂ m₂ m₂ m₁ =1kg m₂ = 2kg k₁ = k₁= k₂ = 1¾ 2₁ Determine (analytically do not use Matlab): 1) The characteristic equation 2) The natural frequencies 3) The mode shapes normalized to the largest value = 1.0 4) Sketch the mode shapes 6) Determine the solutions [10] X₂0 Wis using the modal method. 7) Plot your results using MatLab. 5) Show the mode shapes are orthogonal with respect to the mass matrix x₂ (1) and x₂ (1) for -.7207 rad/s ₁₂ Wn₂ = .1.7552 rad/s 1.0 -0.1754) 11 = 1 [(k₂ + 4k₂ + k₂) −4k₂ 11 -4k₂ 2 = nm 30-4 [0.701 1.0 x₁ (t) = 7.1664 sin(.7207t-.00719)+8.7643 sin 1.1552t+2.9678) x₂ (t)=10.2149 sin(.7207t-.00719)-1.5372 sin 1.1552t +2.9678) {*}={100mm/sec M(t) M(t) R
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
Related questions
Question
Matlab
![### Analytical Determination of System Dynamics
#### Given System Equation:
\[
\begin{bmatrix}
m_1 + \frac{m_2}{2} & -\frac{m_2}{2} \\
-\frac{m_2}{2} & \frac{3}{2}m_2
\end{bmatrix}
\begin{bmatrix}
\ddot{x}_1 \\
\ddot{x}_2
\end{bmatrix}
+
\begin{bmatrix}
c & -c \\
-c & c
\end{bmatrix}
\begin{bmatrix}
\dot{x}_1 \\
\dot{x}_2
\end{bmatrix}
+
\begin{bmatrix}
(k_1 + 4k_2 + k_3) & -4k_2 \\
-4k_2 & 4k_2
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
\frac{-M(t)}{R} \\
\frac{M(t)}{R}
\end{bmatrix}
\]
#### Mass and Stiffness Values:
- \( m_1 = 1 \, \text{kg} \)
- \( m_2 = 2 \, \text{kg} \)
- \( k_1 = k_2 = k_3 = \frac{1}{\sqrt{m}} \)
#### Tasks to Determine:
1. **The Characteristic Equation**
2. **The Natural Frequencies**
3. **The Mode Shapes Normalized to the Largest Value = 1.0**
4. **Sketch the Mode Shapes**
5. **Show Orthogonality of Mode Shapes with Respect to the Mass Matrix**
6. **Compute Solutions \( x_1(t) \) and \( x_2(t) \)** for Initial Conditions:
\[
\begin{bmatrix}
x_{10} \\
x_{20}
\end{bmatrix}
=
\begin{bmatrix}
1 \\
-1
\end{bmatrix}
\, \text{mm}
\quad
\begin{bmatrix}
\dot{x}_{10} \\
\dot{x}_{20}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4b8502c6-77e6-4ad1-93a4-15893327988c%2F3cf77ddb-dcca-4544-9bcc-6de525526e3c%2F96tx35a_processed.png&w=3840&q=75)
Transcribed Image Text:### Analytical Determination of System Dynamics
#### Given System Equation:
\[
\begin{bmatrix}
m_1 + \frac{m_2}{2} & -\frac{m_2}{2} \\
-\frac{m_2}{2} & \frac{3}{2}m_2
\end{bmatrix}
\begin{bmatrix}
\ddot{x}_1 \\
\ddot{x}_2
\end{bmatrix}
+
\begin{bmatrix}
c & -c \\
-c & c
\end{bmatrix}
\begin{bmatrix}
\dot{x}_1 \\
\dot{x}_2
\end{bmatrix}
+
\begin{bmatrix}
(k_1 + 4k_2 + k_3) & -4k_2 \\
-4k_2 & 4k_2
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
\frac{-M(t)}{R} \\
\frac{M(t)}{R}
\end{bmatrix}
\]
#### Mass and Stiffness Values:
- \( m_1 = 1 \, \text{kg} \)
- \( m_2 = 2 \, \text{kg} \)
- \( k_1 = k_2 = k_3 = \frac{1}{\sqrt{m}} \)
#### Tasks to Determine:
1. **The Characteristic Equation**
2. **The Natural Frequencies**
3. **The Mode Shapes Normalized to the Largest Value = 1.0**
4. **Sketch the Mode Shapes**
5. **Show Orthogonality of Mode Shapes with Respect to the Mass Matrix**
6. **Compute Solutions \( x_1(t) \) and \( x_2(t) \)** for Initial Conditions:
\[
\begin{bmatrix}
x_{10} \\
x_{20}
\end{bmatrix}
=
\begin{bmatrix}
1 \\
-1
\end{bmatrix}
\, \text{mm}
\quad
\begin{bmatrix}
\dot{x}_{10} \\
\dot{x}_{20}
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