What Is the Resonance Frequency in a Damped System?

[dchen]

how can i get the resonance frequency for a damped system?
x'' + 2r x' + w. x = F/m coswt
r is the damping factor, w. the natural frequency



it can be calculated that the amplitude,
A=F/m / sqrt( (w.^2-w^2)^2-(2rw)^2)

by differentiating wrt w,
it is found the maximum amplitude occurs at w=sqrt(w.^2-2r^2)
is this the resonance frequency?


or should i look at the frequency w=sqrt(w.^2-r^2), which is the frequency of the transient solution when the system is not forced.
is the solution at this frequency unbounded?
in the form x=Bt cos (sqrt(w.^2-r^2))t

so which is resonance frequency?
w=sqrt(w.^2-2r^2) or w=sqrt(w.^2-r^2) ??

 

[anathema]

Thank you for your question about finding the resonance frequency for a damped system. I am happy to provide some guidance on this topic.

The resonance frequency for a damped system is defined as the frequency at which the system responds with maximum amplitude to an external force. In the equation x'' + 2r x' + w. x = F/m coswt, the natural frequency (w) and damping factor (r) play important roles in determining the resonance frequency.

The equation you provided for calculating the amplitude of the system is correct. However, there are two cases to consider when determining the resonance frequency: underdamped and overdamped.

In the case of an underdamped system (r < w), the maximum amplitude occurs at w=sqrt(w.^2-2r^2). This is the frequency you should use to find the resonance frequency. This is because at this frequency, the system is able to oscillate with maximum amplitude, and the damping factor is not strong enough to prevent this from happening.

In the case of an overdamped system (r > w), the system does not exhibit resonance. Instead, it responds with a steady-state solution, and the maximum amplitude occurs at w=sqrt(w.^2-r^2). This is the frequency you should use to find the maximum amplitude of the system, but it is not considered the resonance frequency.

To summarize, the resonance frequency for an underdamped system is w=sqrt(w.^2-2r^2), while for an overdamped system it is not applicable. I hope this helps clarify the concept of resonance frequency for a damped system. Please let me know if you have any further questions.

 

[m2003]

The resonance frequency in a damped system can be defined as the frequency at which the system experiences maximum amplitude in response to an external force. In this case, the resonance frequency can be calculated using the formula w=sqrt(w.^2-2r^2). This is the frequency at which the amplitude of the system is the highest, indicating a strong response to the external force.

On the other hand, the frequency w=sqrt(w.^2-r^2) represents the frequency of the transient solution when the system is not forced. This solution may not be unbounded, but it does not represent the resonance frequency.

Therefore, the resonance frequency for a damped system is given by w=sqrt(w.^2-2r^2). This frequency can be obtained by differentiating the amplitude formula with respect to w and setting it equal to 0, which gives us w=sqrt(w.^2-2r^2). This frequency is important in understanding the behavior of the damped system and can be used in designing systems to avoid resonance.