The damped harmonic oscillator systemsatisfies the equation mx + rx+ sx = 0.mž +rx +note: pay attention to the xThe displacement experienced by mass mcan be expressed by x = Ae^-rt/2m cos(w't+ Ø ). At t = 0, the mass is at x = x0velocity x (with one point) = v0. On thissystem it is known thatwi – w² = 10-6wž.a. If damping is removed, the mass willocillate in simple harmonic motion.Determine A and Ø in the parameters xo,vo and wo (the natural frequency of theoscillator).b. For this damped harmonic oscillator,determine A and Ø in the parameters xo,vo and w'.c. Calculate the logarithmic reduction9/11and quality factor Q.d. If w, = 10^6 rad/s and m = 10^-10kg,calculate the spring constant s and =coefficient r.

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The damped harmonic oscillator system satisfies the equation mx + rx+ sx = mx + rx + = 0. note: pay attention to the x The displacement experienced by mass m can be expressed by x = Ae^-rt/2m cos(w't + Ø ). At t = 0, the mass is at x = x0 velocity x (with one point) = v0. On this %3D system it is known that w3 – w'² = 10-°wž. a. If damping is removed, the mass will oscillate in simple harmonic motion. Determine A and Ø in the parameters xo, vo and wo (the natural frequency of the oscillator). b. For this damped harmonic oscillator, determine A and Ø in the parameters xo, vo and w'. c. Calculate the logarithmic reduction 9/11 and quality factor Q. d. If w, = 10^6 rad/s and m = 10^-10kg, calculate the spring constant s and = coefficient r.

The damped harmonic oscillator system satisfies the equation mx + rx+ sx = mx + rx + = 0. note: pay attention to the x The displacement experienced by mass m can be expressed by x = Ae^-rt/2m cos(w't + Ø ). At t = 0, the mass is at x = x0 velocity x (with one point) = v0. On this %3D system it is known that w3 – w'² = 10-°wž. a. If damping is removed, the mass will oscillate in simple harmonic motion. Determine A and Ø in the parameters xo, vo and wo (the natural frequency of the oscillator). b. For this damped harmonic oscillator, determine A and Ø in the parameters xo, vo and w'. c. Calculate the logarithmic reduction 9/11 and quality factor Q. d. If w, = 10^6 rad/s and m = 10^-10kg, calculate the spring constant s and = coefficient 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

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Concept explainers

Free Damped Vibrations

Free vibration is a type of vibration in which the external force is removed after giving the initial displacement to a body/system. In other words, in this type of vibration, force is applied at the initial displacement of the system, and after the initial displacement, force is removed, and the system vibrates at the natural frequency.

Question

The damped harmonic oscillator system
satisfies the equation mx + rx+ sx = 0.
mž +rx +
note: pay attention to the x
The displacement experienced by mass m
can be expressed by x = Ae^-rt/2m cos(w't
+ Ø ). At t = 0, the mass is at x = x0
velocity x (with one point) = v0. On this
system it is known that
wi – w² = 10-6wž.
a. If damping is removed, the mass will
ocillate in simple harmonic motion.
Determine A and Ø in the parameters xo,
vo and wo (the natural frequency of the
oscillator).
b. For this damped harmonic oscillator,
determine A and Ø in the parameters xo,
vo and w'.
c. Calculate the logarithmic reduction
9/11
and quality factor Q.
d. If w, = 10^6 rad/s and m = 10^-10kg,
calculate the spring constant s and =
coefficient r.

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