Time constant of a signal

[DrDolittle]

time constant natural response

Can anybody bring a relation between time constant of a signal and the natural response of the signal.........you can also use the example of marching troops on the bridge resulting in its collapse.......i don't need a mathematical treatment,i need a qualitative reasoning

regards
drdolittle

 

[elvis]

relation between pole and time constant

Hi drdolittle

You mean relation between time constant of the system and the frequency response of the system?

Well if this is your question then I think the relation is hidden in the fact that the impulse response and the frequency response are Fourier pair
and because of the fact F{x(a*t)} = 1/abs(a) * X(W/a)

If we have system with pole P then we have corresponding impulse response h(t)=C*exp(P*t)
Pole P impose knee in the frequency response at the pole frequency abs(P)
The time constant based on the equation h(t)=C*exp(P*t); is tal=1/real(P)

Also there is much stronger theory about this. Thy define the with of impulse/frequency response
then find rigorous relation between them ... you can check in the 'Handbook of transforms and Application'

Is this reasoning what you are interested in?


Best Regards
dora

 

[DrDolittle]

The type of relation iam asking for is through characteirstic modes which are nothing but the vlaue of the initial conditions by which the resonance condition can be explained.....try to bridge a relation in these lines.I hope you can understand



regards
drdolittle

 

[ramone]

Hi Drdolittle...

Well if you want someone to answer your question you force him to make some assumptions... Do you mean a system's natural response? propably.... You are talking about one time constant... are you refering to a 1st order system (or at least are you modelling with such an assumption)? i suppose you do that , unless you wouldn't speak for one time constant... Is this pole the dominant??? lets say it is (since you say so)... Well if all above are true then the time constant shows how fast the system terminates its initial conditions... The time constant is at the power of the exponential you have...
About the bridge: i can't really catch what you want to figure out with this but i can surely say that while the Time Constant increases so decreases the effection of the walking troops... Moreover (and that is important) if the signal from the troops (their walk) has the same dominant pole (means the same time constant) with the system (the bridge) then we go out of the stability region and the bridge falls (to see that use a program such as Matlab and force a 1st order system with a signal which its dominant pole is the system's pole... you will see variations with no decreasing width)....

Hope i have helped you... hope i have understood what you have ment

 

[DrDolittle]

I believe you have nicely understoond my question.
My point is this when you have any system you certainly have some initial conditions and these initial conditions are called characteristic roots which will be in exponential to the power lambda.Since these initial conditions have no influence by the input they act on its own and urely subjective to the system being considered.
Thus when u have a system whose input itself is characteristic modes u can find the system to have strrong response(it is like asking an alcoholic to taste liquor.Thus the time constant of a signal has an inverse proportinality with the charcterestic roots as the system's natural response contaons characteristics roots.When the troops marching frequency(i.e,forced response) is made to match with the characteristics roots you will have a resonance phenomenon.

This is another way of explaining it.

Regards
drdolittle