{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

This damping factor is evaluated by a ratio of area

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online