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Damping by mechanism

The damping by mechanism implemented in DeepLines is close to the classical Rayleigh damping except that different damping coefficients may be used for traction/compression, bending and torsion. The equations of motion are assumed uncoupled in the eigenmodes frame of reference and for each mode the system is represented by a one-dimensional oscillator :

\[ m_{\lambda\lambda} \ddot X + c_{\lambda\lambda} \dot X + k_{\lambda\lambda} X = F_{\lambda} \]

For a given mechanism defined by its natural frequency \(\omega_{\lambda} = \sqrt {\frac {k_{\lambda\lambda}}{m_{\lambda\lambda}}}\) , a damping rate \(\xi\) may be introduced :

\[ \ddot X + 2\xi \omega _{\lambda} \dot X +\omega _{\lambda}^2 X = \frac{F_{\lambda} }{m_{\lambda\lambda}} \]

with \(\xi = \frac {C_{\lambda\lambda}}{C_{crit \lambda}} = \frac {\beta \omega_{\lambda}}{2}\).

Warning

The damping coefficients used during your dynamic analyses may be defined either as beta or actual damping rate ksi. It is important to note that when using a beta type coefficient, the actual damping rate will depend on the motion's period.

It is important to mention that for a given damping rate, the \(\beta\) coefficient depends on which structural natural mode is damped. The solution of the equation of motion writes :

\[ X = X_0 + \Big \{Acos(\sqrt {1-\xi^2 }\omega t) + B sin(\sqrt {1-\xi^2 }\omega t) \Big \} exp(-\xi \omega t) \]

which gives a time series with the following classical shape :

The damping ratio may be directly calculated from the curve :

\[ \xi + \theta (\xi_2) = \frac {ln(K_n)}{2\pi} \text { with } K_n = \frac {X_n - X_{n-1}}{X_{n+1}-X_n} \]