Proportionally damped 2-DOF system

[toaster89]

A proportionally damped 2-DOF system has mass and stiffness matrix M and K. We also know that the system has damping ratio ζ1 = 0.1 and 2 = 0.3. The damping matrix is written as
C = α M + β K + γ KM^-1K
Try to find the coefficients

Mx"+Cx'+kx=0
CM^-1K=KM^-1C

Mx"(t)+(αM+βK+γKM^-1K)x'(t)+kx(t)=0
x(t)=M^-.5 x q and multiply by M^-.5

q"(t) + (αI+βK+γM^-.5 M^-.5 KM^-1K)q'(t)+Kq(t)=0

q"(t)+(αI+βK+γ2M^-.5K)q't+Kq(t)=0


So far this is all i got and now i am stuck

any tips?

 

[bluechair]

I'm trying to figure out the same problem, were you ever able to figure it out?

 

[AlephZero]

I don't know why you have 3 unknown coefficients in your expression for C when you only have two requirements to meet. That will not give you a unique answer. You might as well assume γ = 0.

C = α M gives a damping ratio proportional to 1/ω.
C = β K gives a damping ratio proportional to ω.
The mode shapes with Rayleigh damping are the same as with no damping.

(Those results should be in your textbook or course notes).

So you get two simultaneous equations to find α and β in terms of ζ1 and ζ2.