According to Wikipedia, the Q factor is defined as:

$$Q=2\pi\frac{\mathrm{energy \, \, stored}}{\mathrm{energy \, \,dissipated \, \, per \, \, cycle}}$$

Here are my questions:

1. Does the energy dissipated per cycle assume that the amplitude is constant from one cycle to the next.
2. Is it always calculated at the resonance frequency?
3. If the answer to 2 is yes can you explain why for a forced oscillator system with a damping coefficient of $\gamma$ and natural frequency $\omega_o$ the quality factor is $Q=\omega_o/\gamma$ and not some more complicated expression involving the actual resonant frequency (which is not quite $\omega_o$ and is given by $$\omega_r=(\omega_0^2-\frac{\gamma^2}{2})^{\frac{1}{2}}$$ of the system? Is this just an approximation i.e. are we assuming that it resonates at $\omega_o$ but in fact the actual expression is a big more complicated?

Edit: With using the actual value of $\omega_r$ I get: $Q=(\omega_0^2-\frac{\gamma^2}{2})^{\frac{1}{2}}/\gamma$ is this more correct?

According to Wikipedia, the $Q$ factor is defined as:

$$Q=2\pi\frac{\mathrm{energy \, \, stored}}{\mathrm{energy \, \,dissipated \, \, per \, \, cycle}}.$$

Here are my questions:

1. Does the energy dissipated per cycle assume that the amplitude is constant from one cycle to the next.
2. Is it always calculated at the resonance frequency?
3. If the answer to 2 is yes can you explain why for a forced oscillator system with a damping coefficient of $\gamma$ and natural frequency $\omega_o$ the quality factor is $Q=\omega_o/\gamma$ and not some more complicated expression involving the actual resonant frequency (which is not quite $\omega_o$ and is given by $$\omega_r=(\omega_0^2-\frac{\gamma^2}{2})^{\frac{1}{2}}$$ of the system? Is this just an approximation i.e. are we assuming that it resonates at $\omega_o$ but in fact the actual expression is a big more complicated?

Edit: With using the actual value of $\omega_r$ I get: $Q=(\omega_0^2-\frac{\gamma^2}{2})^{\frac{1}{2}}/\gamma$ is this more correct?

According to Wikipedia, the Q factor is defined as:

$$Q=2\pi\frac{\mathrm{energy \, \, stored}}{\mathrm{energy \, \,dissipated \, \, per \, \, cycle}}$$

Here are my questions:

1. Does the energy dissipated per cycle assume that the amplitude is constant from one cycle to the next.
2. Is it always calculated at the resonance frequency?
3. If the answer to 2 is yes can you explain why for a forced oscillator system with a damping coefficient of $\gamma$ and natural frequency $\omega_o$ the quality factor is $Q=\omega_o/\gamma$ and not some more complicated expression involving the actual resonant frequency (which is not quite $\omega_o$) of the system? Is this just an approximation i.e. are we assuming that it resonates at $\omega_o$ but in fact the actual expression is a lot more complicated?

According to Wikipedia, the Q factor is defined as:

$$Q=2\pi\frac{\mathrm{energy \, \, stored}}{\mathrm{energy \, \,dissipated \, \, per \, \, cycle}}$$

Here are my questions:

1. Does the energy dissipated per cycle assume that the amplitude is constant from one cycle to the next.
2. Is it always calculated at the resonance frequency?
3. If the answer to 2 is yes can you explain why for a forced oscillator system with a damping coefficient of $\gamma$ and natural frequency $\omega_o$ the quality factor is $Q=\omega_o/\gamma$ and not some more complicated expression involving the actual resonant frequency (which is not quite $\omega_o$ and is given by $$\omega_r=(\omega_0^2-\frac{\gamma^2}{2})^{\frac{1}{2}}$$ of the system? Is this just an approximation i.e. are we assuming that it resonates at $\omega_o$ but in fact the actual expression is a big more complicated?

Edit: With using the actual value of $\omega_r$ I get: $Q=(\omega_0^2-\frac{\gamma^2}{2})^{\frac{1}{2}}/\gamma$ is this more correct?