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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.
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
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:
&& + 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
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
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
K (3-20) Accordingly, the eignvalu...
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This note was uploaded on 03/14/2014 for the course CE 5680 taught by Professor Drgrd during the Summer '14 term at Indian Institute of Technology, Chennai.
- Summer '14