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(adsbygoogle = window.adsbygoogle || []).push({});

C = α M + β K + γ KM^{-1}K

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^{-1}K)q'(t)+Kq(t)=0

q"(t)+(αI+βK+γ2M^{-.5}K)q't+Kq(t)=0

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

any tips?

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# Proportionally damped 2-DOF system

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