What Are the Correct Formulations for the Damping Coefficient?

[smokedvanilla]

Hi, I have been looking for formulae for the damping coefficient, and I found two different formulae for it.

http://www.xyobalancer.com/xyo-balancer-blog/viscous-damping-coefficient
This webpage states that damping force, Fd is given by Fd=-cv, where c is the damping coefficient while v is the velocity.

http://hyperphysics.phy-astr.gsu.edu/hbase/oscda.html
However, this page states that damping coefficient is given by c/2m, where c=-Fd/v

Is there a fixed definition for damping coefficient, or are both definitions acceptable?

 

[Hesch]

Is there a fixed definition for damping coefficient, or are both definitions acceptable?

No, they are different:
In your first link, it's stated that damping ratio = c.
The second link states that damping ratio = c/2m.

I think there is some confusion as to terminology, but I prefer the second link:

The last equation is the characteristic equation of the system, which could be written by Lapace:

m * s² + c * s + k = 0 ⇒

s² + (c/m)s + k/m = 0

The last equation could also be written:

s² + 2ξω_ns + ω_n² = 0

where ξ is the damping ratio in my definition, which regards the ratio that the amplitude of some oscillation has been reduced from one period to the following:

Amplitude_n+1 = Amplitude_n * ( 1 - ξ )

 

[smokedvanilla]

First of all, thank you for your reply.

s2 + 2ξωns + ωn2 = 0

where ξ is the damping ratio in my definition, which regards the ratio that the amplitude of some oscillation has been reduced from one period to the following.

Which gives ξ=c/(2*√mk), so c is the damping coefficient & 2*√mk is the damping coefficient in the case of critical damping?

 

[Hesch]

Which gives ξ=c/(2*√mk) . . . . .

No,

ξ = c / ( 2m * ω_n ) , ω_n = √( k/m )

Try again, one step at a time.

 

[Hesch]

Amplituden+1 = Amplituden * ( 1 - ξ )

Correction:
This is not right, but the connection can be seen here:

 

[smokedvanilla]

The ga

Correction:
This is not right, but the connection can be seen here:

is the graph wrong because zeta should be 1 instead of 0.4 when critical damping occurs? I think I understood the concept.

 

[Hesch]

is the graph wrong because zeta should be 1 instead of 0.4 when critical damping occurs?

No, I trust the graphs, but my expression in #2:

Amplitude_n+1 = Amplitude_n * ( 1 - ξ )

was simply wrong.

Maybe you could calculate the correct expression? ( 1 - ξ ) must be substituted by something else ( f(ξ) ).

Critical damping is when ξ = 1. No damping is when ξ = 0. In "typical" analog controllers as for temperature, motorspeed, a ξ = 0.65 . . 0.7 is often chosen, which compromises control speed and overshoot. In digital controllers a ξ = 1 is often chosen.

 

[smokedvanilla]

No, I trust the graphs, but my expression in #2:

Amplitude_n+1 = Amplitude_n * ( 1 - ξ )

was simply wrong.

Maybe you could calculate the correct expression? ( 1 - ξ ) must be substituted by something else ( f(ξ) ).

Critical damping is when ξ = 1. No damping is when ξ = 0. In "typical" analog controllers as for temperature, motorspeed, a ξ = 0.65 . . 0.7 is often chosen, which compromises control speed and overshoot. In digital controllers a ξ = 1 is often chosen.

Thank you about the graph, I understand that now :D

I tried working the amplitude expression out,
https://www.google.com/search?q=damping+ratio+formula&es_sm=91&source=lnms&tbm=isch&sa=X&ved=0CAcQ_AUoAWoVChMIn-qy07aPxwIV1Y-OCh2PVwZU&biw=1277&bih=637#imgrc=ArC96HvFuNB9TM%3A
Since ξ is the ratio of damping coefficient of the system when critical damping occurs (Cc) to the damping coefficient of the oscillation (γ), ξ=γ/Cc. From this we have γ=ξ*Cc--(1)

The expression for variation of amplitude with time is Ae^(-γt), so if we sub. (1) into the expression we get Amplitude_n+1 = Amplitude_n*e^(-ξ*Cc)t

 

[Hesch]

if we sub. (1) into the expression we get Amplitude_n+1 = Amplitude_n*e^(-ξ*Cc)t

Yes, that is right: If we damp an oscillation ( say sin(ωt) ) the sin(ωt)-function will be enveloped within a ±exponential-function. But if you look closely at the graphs in #6, you will see that the damped sin(ωt) has a somewhat lower frequency than the undamped sin(ωt): Thus the amplitude (peakvalue) will be somewhat smaller than
Amplitude_n+1 = Amplitude_n*e^(-ξ*Cc)t.

The ξ could be formulated as:

Say you have a characteristic equation: s² + 2s + 2 = 0 , you will find the roots z=( -1 + j1 ) and z=( -1 - j1 ). If you plot these roots in a root locus, and draw two lines through ( 0 , 0 ) and ( -1 ± j1 ) we could call the angles between these lines and the imaginary axis φ ( = 45° ). Then ξ = sin(φ) = 0.7071. So if you want a ξ = 0.6 , the roots of the characteristic equation must be on two lines with an angle = arcsin( 0.6 ) as to the imaginary axis ( = 36.87° ).

Critical damping is exactly when the roots close up on the real axis ( φ = 90° , ξ = 1 ):