Quality Factor in damped oscillation

In summary: My problem being that I cannot prove that \frac{b}{2m\omega_{0}} = \frac{1}{4Q^{2}} Knowing that E = E_{0}exp^{-bt/m} i tried finding the energy loss per cycle by finding the difference between the energy at time t and the energy at time t + T (where T is the time period) but just ened up with an unhelpfull equation.\frac{1}{4Q^{2}}} is the loss in energy per cycle when the Q factor is greater than 1/2.
  • #1
Mattofix
138
0
Working through my lecture summaries, I have been given that [tex] Q (the quality factor) =\frac{2\pi}{(\Delta E/E)cycle}[/tex]

and accepted this as a statement, taking [tex]\((\Delta E/E)cycle}[/tex] to mean the 'energy loss per cycle'.

The notes carry on to say

'The frequency [tex]\widetilde{\omega}[/tex] of under(damped) oscillator as function of the frequency [tex]\omega_{0}[/tex] and the Q factor:

[tex]\widetilde{\omega} = \omega_{0}\sqrt{1 - (\frac{b}{2m\omega_{0})^{2}}} = \omega_{0}\sqrt{1 - \frac{1}{4Q^{2}}[/tex]

My problem being that I cannot prove that [tex]\frac{b}{2m\omega_{0}} = \frac{1}{4Q^{2}} [/tex]

Knowing that [tex]E = E_{0}exp^{-bt/m}[/tex] i tried finding the energy loss per cycle by finding the difference between the energy at time t and the energy at time t + T (where T is the time period) but just ened up with an unhelpfull equation.
 
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  • #2
any help would be much appreciated so i can get rid of this irritating missing link.
 
  • #3
ok - i appreciate that [tex]
\((\Delta E/E)cycle}
[/tex] means energy loss per cycle divided by energy stored - where energy stored would be [tex]
E = E_{0}exp^{-bt/m}
[/tex]

but i still cannot prove it
 
  • #4
Mattofix said:
Working through my lecture summaries, I have been given that [tex] Q (the quality factor) =\frac{2\pi}{(\Delta E/E)cycle}[/tex]

and accepted this as a statement, taking [tex]\((\Delta E/E)cycle}[/tex] to mean the 'energy loss per cycle'.

The notes carry on to say

'The frequency [tex]\widetilde{\omega}[/tex] of under(damped) oscillator as function of the frequency [tex]\omega_{0}[/tex] and the Q factor:

[tex]\widetilde{\omega} = \omega_{0}\sqrt{1 - (\frac{b}{2m\omega_{0})^{2}}} = \omega_{0}\sqrt{1 - \frac{1}{4Q^{2}}[/tex]

My problem being that I cannot prove that [tex]\frac{b}{2m\omega_{0}} = \frac{1}{4Q^{2}} [/tex]

Knowing that [tex]E = E_{0}exp^{-bt/m}[/tex] i tried finding the energy loss per cycle by finding the difference between the energy at time t and the energy at time t + T (where T is the time period) but just ened up with an unhelpfull equation.

i can tell you why, if [itex]Q>\sqrt{1/2}[/itex] that the peak resonant frequency is

[tex]\omega_{0}\sqrt{1 - \frac{1}{4Q^{2}}[/tex]

if [itex]\omega_0[/itex] the "natural" resonant frequency (i don't know what to call it) of the system. but i do not know what b and m are and can't tell from the context. is this a second order mechanical system or an electrical system?
 

1. What is the definition of Quality Factor (Q) in damped oscillation?

The Quality Factor, also known as Q-factor, is a measure of the rate at which energy is lost in a damped oscillating system. It is defined as the ratio of the energy stored in the system to the energy dissipated per cycle.

2. How does the value of Q affect the behavior of a damped oscillating system?

The higher the value of Q, the less energy is lost per cycle and the longer the system will continue to oscillate. A lower Q value indicates a greater rate of energy dissipation, resulting in shorter oscillation periods and a more rapid decay of the oscillations.

3. What factors influence the Q-value of a damped oscillation system?

The Q-value is influenced by the damping coefficient, the mass of the system, and the stiffness of the restoring force. In general, a higher damping coefficient and a lower mass or stiffness will result in a lower Q-value.

4. How is the Q-factor calculated in a damped oscillation system?

The Q-factor can be calculated using the formula Q = 2πE/Ed, where E is the energy stored in the system and Ed is the energy dissipated per cycle. Alternatively, Q can also be calculated using the equation Q = ω0/2β, where ω0 is the natural frequency of the system and β is the damping ratio.

5. What is the significance of Q-factor in practical applications?

The Q-factor is an important parameter in many practical applications, such as in designing electronic circuits, tuning musical instruments, and studying the behavior of mechanical systems. It also plays a crucial role in determining the stability and accuracy of oscillating systems.

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