Neper frequency, damped harmonic oscillation

[FrankJ777]

Hi all
I'm not sure if this question is better suited for the EE thread or diff eq, but I'm trying to understand what the neper frequency, [tex]\alpha[/tex], signifies. I know it's supposed to be the damping factor and that its units are rad/second, but I'm not sure what that implies. It would seem to indicate, by its units, that its rate of oscillation slows by [tex]\alpha[/tex] [tex]rad/second[/tex], but of course period and frequency remain constant durring damped harmonic oscillation. So can anyone explain to me what I'm missing. In other words if I have a RCL circuit, or a mass and spring system for that matter, what can I predict about the oscillation knowing [tex]\alpha[/tex]?

Thanks a lot.

 

[Bob S]

A neper is the natural scale of attenuation, and equals 1/e = 0.367879 = -8.686 decibels.
So e^-αt is attenuation of αt nepers.

α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ ς σ τ υ φ χ ψ ω .

 

[oswaler]

I can provide some insight into the concept of Neper frequency and its significance in damped harmonic oscillation. The Neper frequency, also known as the damping frequency or attenuation factor, is a measure of how quickly the amplitude of a damped harmonic oscillator decreases over time. It is represented by the symbol \alpha and has units of radians per second.

In a damped harmonic oscillation system, there are three important factors at play: the natural frequency, the damping coefficient, and the driving force. The natural frequency is the frequency at which the system would oscillate if there were no external forces acting on it. The damping coefficient is a measure of the resistance to motion in the system, and the driving force is any external force that is causing the oscillation.

Now, the Neper frequency is directly related to the damping coefficient. It represents the rate at which the energy of the system is dissipated due to the damping. In other words, as the system oscillates, the damping force acts to decrease the amplitude of the oscillation, and the Neper frequency determines how quickly this occurs.

In terms of predicting the behavior of a damped harmonic oscillator, knowing the Neper frequency can tell us a few things. Firstly, a higher Neper frequency means a higher damping coefficient, which results in a faster decrease in amplitude. This means that the oscillation will reach equilibrium (where the amplitude becomes zero) more quickly.

Additionally, the Neper frequency can also tell us something about the stability of the system. In a damped harmonic oscillator, there is a critical damping coefficient at which the system is in a state of critical damping and will return to equilibrium in the shortest possible time. The Neper frequency can be used to calculate this critical damping coefficient, providing valuable information about the stability of the system.

In summary, the Neper frequency is an important parameter in understanding the behavior of damped harmonic oscillation systems. It represents the rate at which energy is dissipated due to damping and can provide valuable insights into the stability and behavior of the system.