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E0-190-2008_(4)Chapter_3

# This damping factor is evaluated by a ratio of area

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Unformatted text preview: re. This damping factor is evaluated by a ratio of area of a hysterics loop to an elastic restoring energy. P ΔW We δ We : Elastic Potential Energy ΔW: Loop Area Damping Factor: h=ΔW/(2πWe) As same as the damping factor aforementioned, we can estimate a damping factor of D.I.F. Let consider how to estimate it. A equation of motion of a single degree of freedom system during the free vibration is expressed by: & m&& + cx + kx = 0 x (3-10) where, m=mass, c=coefficient of viscous damping and k= spring constant. Introducing an undamped natural circular frequency ωo(=√k/m) and a damping factor h(=cωo/2k) , Eq.(3-10) can be rewritten as: 2 && + 2hωo x + ωo x = 0 & x (3-11) Putting x=Xexp(λt) and substituting into Eq.(3-11), the following eigenvalue equation is obtained. (3-12) Solving a quadratic equation of Eq.(3-12), the eigenvalue λis obtained as: λ = −hωo + iωo 1 − h2 (3-13) Express the real and the imaginary part of the eigenvalue λ by: Re .(λ ) = −hωo (3-14) Im .( λ ) = ωo 1 − h2 (3-15) and calculate η which is defined by: η= − Re .( λ ) (Re .( λ ))2 + (Im .( λ ))2 =h (3-16) In a case where a single degree of freedom system has a spring of complex number type such as D.I.F, an equation of free vibration is expressed by: (3-17) Where, K+iK’ is the complex spring constant. The eignvalue equation for Eq.(3-17) is: mλ2 + (K + iK ′) = mλ2 + K * exp(i2φ) = 0 where, (3-18) K * = K 2 + K ′2 (3-19) 1 −1( K ′ ) φ = tan 2 K (3-20) Accordingly, the eignvalu...
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