Find Resonant Frequency of Spring-Mass-Dashpot System

[Poetria]

Homework Statement

Consider the spring-mass-dashpot system driven by a sinusoidal force on the mass:

(mD2+bD+k)x=F where sin(omega*t)

Recall that for the unforced underdamped oscillator (b^2<4km, F=0), the value of the natural damped frequency omega_d=sqrt(k/m - b^2/(4m^2)
Find the resonant (angular) frequency omega_r.

That is, find the angular frequency at which the gain of the response attains its maximum.

(This resonant frequency will be in the form sqrt(H) for some expression H. Assume H>0)
----
f(omega)=(k-m*omega^2)^2+b^2*omega^2
f'(omega=-4*m*omega*(k-m*omega^2)+2*b^2*omega=0

I got this one right - sqrt(k/m-b^2/(2*m^2)) :)
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Now I have a problem:
Finally, what happens if the expression H is negative or zero? Find the angular frequency omega at which the gain attains its maximum for the case when H is less or equal to 0.

Switching signs does not help. :(

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[Diegor]

Homework Statement

Consider the spring-mass-dashpot system driven by a sinusoidal force on the mass:

(mD2+bD+k)x=F where sin(omega*t)

Recall that for the unforced underdamped oscillator (b^2<4km, F=0), the value of the natural damped frequency omega_d=sqrt(k/m - b^2/(4m^2)
Find the resonant (angular) frequency omega_r.

That is, find the angular frequency at which the gain of the response attains its maximum.

(This resonant frequency will be in the form sqrt(H) for some expression H. Assume H>0)
----
f(omega)=(k-m*omega^2)^2+b^2*omega^2
f'(omega=-4*m*omega*(k-m*omega^2)+2*b^2*omega=0

I got this one right - sqrt(k/m-b^2/(2*m^2)) :)
---
Now I have a problem:
Finally, what happens if the expression H is negative or zero? Find the angular frequency omega at which the gain attains its maximum for the case when H is less or equal to 0.

Switching signs does not help. :(

----

Your H is negative if b is big enough but that would mean that the damping is so big that the system won't have any resonant frecuency because there is no chance that it naturally oscilates( it means With F=0)

 

[Poetria]

Your H is negative if b is big enough but that would mean that the damping is so big that the system won't have any resonant frecuency because there is no chance that it naturally oscilates( it means With F=0)

Yes, I do know it but how to write the equation? I thought it would be sqrt(k/m-b^2/(4*m^2)) but it is wrong. :( If I assume that
-(k-m*omega^2)^2+b^2*omega^2) I get the same result through differentiation.

 

[Poetria]

Ok. Many thanks. I got it. :)