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MODAL_DAMPING.html
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<html><head><link rel="stylesheet" type="text/css" href="style.css"/></head><body> <H2> <BR> *MODAL DAMPING </H2> <P> Keyword type: step <P> This card is used within a step in which the *MODAL DYNAMIC or *STEADY STATE DYNAMICS procedure has been selected. There are two optional, mutually exclusive parameters: RAYLEIGH and MODAL=DIRECT (default). <P> If MODAL=DIRECT is selected the user can specify the viscous damping factor <SPAN CLASS="MATH"><B><IMG WIDTH="12" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img345.png" ALT="$ \zeta$"></B></SPAN> for each mode separately. This is the default. Direct damping is not allowed in combination with nonzero single point constraints in a *MODAL DYNAMIC procedure (however, in a *STEADY STATE DYNAMICS procedure it is). <P> If RAYLEIGH is selected Rayleigh damping is applied in a global way, i.e. the damping matrix <!-- MATH $\left [ C \right ]$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="25" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img2308.png" ALT="$ \left [ C \right ]$"></B></SPAN> is taken to be a linear combination of the stiffness matrix <!-- MATH $\left [ K \right ]$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img2309.png" ALT="$ \left [ K \right ]$"></B></SPAN> and the mass matrix <!-- MATH $\left [ M \right ]$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="30" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img2310.png" ALT="$ \left [ M \right ]$"></B></SPAN>: <P> <P></P> <DIV ALIGN="CENTER" CLASS="mathdisplay"><!-- MATH \begin{equation} \left [ C \right ] = \alpha \left [ M \right ] + \beta \left [ K \right ]. \end{equation} --> <TABLE CLASS="equation" CELLPADDING="0" WIDTH="100%" ALIGN="CENTER"> <TR VALIGN="MIDDLE"> <TD NOWRAP ALIGN="CENTER"><SPAN CLASS="MATH"><IMG WIDTH="148" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img2311.png" ALT="$\displaystyle \left [ C \right ] = \alpha \left [ M \right ] + \beta \left [ K \right ].$"></SPAN></TD> <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> (<SPAN CLASS="arabic">724</SPAN>)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> <P> The coefficients apply to all modes. The corresponding viscous damping factor <SPAN CLASS="MATH"><B><IMG WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2471.png" ALT="$ \zeta_j$"></B></SPAN> for mode j amounts to: <P> <P></P> <DIV ALIGN="CENTER" CLASS="mathdisplay"><!-- MATH \begin{equation} \zeta_j=\frac{\alpha}{2 \omega_j}+ \frac{\beta \omega_j}{2}. \end{equation} --> <TABLE CLASS="equation" CELLPADDING="0" WIDTH="100%" ALIGN="CENTER"> <TR VALIGN="MIDDLE"> <TD NOWRAP ALIGN="CENTER"><SPAN CLASS="MATH"><IMG WIDTH="121" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" SRC="img2472.png" ALT="$\displaystyle \zeta_j=\frac{\alpha}{2 \omega_j}+ \frac{\beta \omega_j}{2}.$"></SPAN></TD> <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> (<SPAN CLASS="arabic">725</SPAN>)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> <P> Consequently, <SPAN CLASS="MATH"><B><IMG WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" SRC="img138.png" ALT="$ \alpha$"></B></SPAN> damps the low frequencies, <SPAN CLASS="MATH"><B><IMG WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img561.png" ALT="$ \beta$"></B></SPAN> damps the high frequencies. <P> The *MODAL DAMPING keyword can be used in any step to redefine damping values defined in a previous step. <P> <P><P> <BR> <P> First line: <UL> <LI>*MODAL DAMPING,RAYLEIGH </LI> <LI>Enter any needed parameters and their values. </LI> </UL> <P> Second line if MODAL=DIRECT is selected (or, since this is default, if no additional parameter is entered): <UL> <LI>lowest mode of the range </LI> <LI>highest mode of the range (default is lowest mode of the range) </LI> <LI>viscous damping factor <SPAN CLASS="MATH"><B><IMG WIDTH="12" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img345.png" ALT="$ \zeta$"></B></SPAN> for modes between (and including) the lowest and highest mode of the range </LI> </UL> Repeat this line if needed. <P> Second line if RAYLEIGH is selected: <UL> <LI>not used (kept for compatibility reasons with ABAQUS) </LI> <LI>not used (kept for compatibility reasons with ABAQUS) </LI> <LI>Coefficient of the mass matrix <SPAN CLASS="MATH"><B><IMG WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" SRC="img138.png" ALT="$ \alpha$"></B></SPAN>. </LI> <LI>Coefficient of the stiffness matrix <SPAN CLASS="MATH"><B><IMG WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img561.png" ALT="$ \beta$"></B></SPAN>. </LI> </UL> <P> <PRE>
Example:
*MODAL DAMPING,RAYLEIGH
,,0.,2.e-4
</PRE> <P> indicates that the damping matrix is obtained by multiplying the stiffness matrix with <!-- MATH $2 \cdot 10^{-4}$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="56" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="img2320.png" ALT="$ 2 \cdot 10^{-4}$"></B></SPAN> <P> <P><P> <BR> Example files: beamdy3, beamdy4, beamdy5, beamdy6. <P> </body></html>