M 1 {[1 '* . _ (w/wn)²]² + [2¢w/wn] 1/2 2 1²}' ||
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Solve for zeta
![The displayed formula represents the calculation for the Magnification Factor (M) in a damped harmonic oscillator. It is expressed as:
\[
M = \frac{1}{\left\{ \left[ 1 - (\omega / \omega_n)^2 \right]^2 + \left[ 2 \zeta \omega / \omega_n \right]^2 \right\}^{1/2}}
\]
Where:
- \( \omega \) is the frequency of excitation.
- \( \omega_n \) is the natural frequency of the system.
- \( \zeta \) is the damping ratio.
This equation is used to quantify how much the amplitude of a system's response is magnified at different frequencies of excitation relative to its natural frequency. The formula incorporates both the relationship between the excitation and natural frequencies and the influence of damping on the system's behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F811e9e09-f827-40da-a830-8949fa4f6658%2F2d03be78-711f-4e1f-93ab-b07d0f98d3b5%2Frffr293_processed.png&w=3840&q=75)
Transcribed Image Text:The displayed formula represents the calculation for the Magnification Factor (M) in a damped harmonic oscillator. It is expressed as:
\[
M = \frac{1}{\left\{ \left[ 1 - (\omega / \omega_n)^2 \right]^2 + \left[ 2 \zeta \omega / \omega_n \right]^2 \right\}^{1/2}}
\]
Where:
- \( \omega \) is the frequency of excitation.
- \( \omega_n \) is the natural frequency of the system.
- \( \zeta \) is the damping ratio.
This equation is used to quantify how much the amplitude of a system's response is magnified at different frequencies of excitation relative to its natural frequency. The formula incorporates both the relationship between the excitation and natural frequencies and the influence of damping on the system's behavior.
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