Understanding Resonance in Underdamped Forced Vibrations

[HimanshuM2376]

What is the condition for resonance to occur in case of underdamped forced vibrations?

 

[anorlunda]

lack of negative feedback

 

[berkeman]

And the frequency of the vibrations = ______ ?

 

[HimanshuM2376]

lack of negative feedback

Can you please elaborate?

 

[anorlunda]

Can you please elaborate?

Negative feedback causes damping. Less feedback, less damping. The tricky part comes with the phase relationship of the feedback. It is bet described with mathematics, data and graphics.

Try reading this article.
https://en.m.wikipedia.org/wiki/Resonance

 

[SCP]

Resonance occurs when the input to a system occurs at a frequency that matches a natural frequency of the system. When this happens, the input continuously adds energy to the system, so oscillations get continuously larger. In a simplified mathematical model of an undamped system, the amplitude of the system output will go to infinity during resonance. In the real world, either system failure (for example a broken spring in a mechanical system), non-linearities (such as the spring stiffness changing as it flexes), or the presence of damping (such as friction in mechanical systems) will limit the resonant amplitude to some finite value.

 

[HimanshuM2376]

Resonance occurs when the input to a system occurs at a frequency that matches a natural frequency of the system. When this happens, the input continuously adds energy to the system, so oscillations get continuously larger. In a simplified mathematical model of an undamped system, the amplitude of the system output will go to infinity during resonance. In the real world, either system failure (for example a broken spring in a mechanical system), non-linearities (such as the spring stiffness changing as it flexes), or the presence of damping (such as friction in mechanical systems) will limit the resonant amplitude to some finite value.

Thanks. What I want to know is that for an underdamped system undergoing forced vibration the maximum amplitude occurs when the excitation frequency is less than natural frequency when we increase the value of damping ratio.

 

[SCP]

That's correct. There is a distinction between the undamped natural frequency ($\omega _n$) and the damped natural frequency ($\omega _d$). In terms of language, when someone says "natural frequency", they usually mean $\omega _n$. Damping is usually expressed in terms of the damping ratio, $\zeta$ (zeta). For an underdamped ($\zeta < 1$) linear system with 1 degree of freedom, the relationship between the two is $\omega _d = \omega _n\sqrt{1-\zeta ^2}$. So as $\zeta$ increases toward 1, $\omega _d$ decreases toward 0.

 

[tech99]

Thanks. What I want to know is that for an underdamped system undergoing forced vibration the maximum amplitude occurs when the excitation frequency is less than natural frequency when we increase the value of damping ratio.

I believe this is equivalent to the electrical analogy of a parallel resonant circuit. Resonance is sometimes defined as the frequency when current and voltage are in-phase. But for the heavily damped parallel circuit, this frequency does not coincide with maximum amplitude.