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DAMPING.html
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DAMPING.html
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<html><head><link rel="stylesheet" type="text/css" href="style.css"/></head><body> <H2> <BR> *DAMPING </H2> <P> Keyword type: model definition, if structural damping: material <P> This card is used to define Rayleigh damping for implicit and explicit dynamic calculations (*DYNAMIC) and structural damping for steady state dynamics calculations (*STEADY STATE DYNAMICS). <P> For Rayleigh damping there are two required parameters: ALPHA and BETA. <P> 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 ].$"></SPAN></TD> <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> (<SPAN CLASS="arabic">697</SPAN>)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> <P> The damping force satisfies: <P> <P></P> <DIV ALIGN="CENTER" CLASS="mathdisplay"><!-- MATH \begin{equation} \lbrace F \rbrace = \left [ C \right ] \lbrace v \rbrace, \end{equation} --> <TABLE CLASS="equation" CELLPADDING="0" WIDTH="100%" ALIGN="CENTER"> <TR VALIGN="MIDDLE"> <TD NOWRAP ALIGN="CENTER"><SPAN CLASS="MATH"><IMG WIDTH="106" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img2312.png" ALT="$\displaystyle \lbrace F \rbrace = \left [ C \right ] \lbrace v \rbrace,$"></SPAN></TD> <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> (<SPAN CLASS="arabic">698</SPAN>)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> <P> where <!-- MATH $\lbrace v \rbrace$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="28" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img2313.png" ALT="$ \lbrace v \rbrace$"></B></SPAN> is the velocity vector. For Rayleigh damping only one *DAMPING card can be used in the input deck. It applies to the whole model. <P> For explicit dynamic calculations only mass proportional damping is allowed, i.e. <SPAN CLASS="MATH"><B><IMG WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img561.png" ALT="$ \beta$"></B></SPAN> must be zero. <P> For structural damping the damping is a material characteristic. Each material can have its own damping value. There is one required parameter STRUCTURAL, defining the value <SPAN CLASS="MATH"><B><IMG WIDTH="12" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img345.png" ALT="$ \zeta$"></B></SPAN> of the damping. For structural damping the element damping force is displacement dependent and satisfies: <P> <P></P> <DIV ALIGN="CENTER" CLASS="mathdisplay"><!-- MATH \begin{equation} \lbrace F \rbrace_e = i \zeta_e \left [ K \right ]_e \lbrace x \rbrace_e, \end{equation} --> <TABLE CLASS="equation" CELLPADDING="0" WIDTH="100%" ALIGN="CENTER"> <TR VALIGN="MIDDLE"> <TD NOWRAP ALIGN="CENTER"><SPAN CLASS="MATH"><IMG WIDTH="151" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img2314.png" ALT="$\displaystyle \lbrace F \rbrace_e = i \zeta_e \left [ K \right ]_e \lbrace x \rbrace_e,$"></SPAN></TD> <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> (<SPAN CLASS="arabic">699</SPAN>)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> <P> where <!-- MATH $i=\sqrt{-1}$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0" SRC="img2315.png" ALT="$ i=\sqrt{-1}$"></B></SPAN>, <SPAN CLASS="MATH"><B><IMG WIDTH="34" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img2316.png" ALT="$ [K]_e$"></B></SPAN> is the element stiffness matrix, and <!-- MATH $\lbrace x \rbrace_e$ --> <SPAN CLASS="MATH"><B><IMG WIDTH="36" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img2317.png" ALT="$ \lbrace x \rbrace_e$"></B></SPAN> is the element displacement vector. <SPAN CLASS="MATH"><B><IMG WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="img2318.png" ALT="$ \zeta_e$"></B></SPAN> is the structural damping value for the material of element <SPAN CLASS="MATH"><B><IMG WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" SRC="img2129.png" ALT="$ e$"></B></SPAN> (default is zero). The global damping force is assembled from the element damping forces. <P> <P><P> <BR> <P> First line: <UL> <LI>*DAMPING </LI> <LI>Enter ALPHA and BETA and their values for Rayleigh damping or STRUCTURAL and its value for structural damping. </LI> </UL> <P> <PRE>
Example:
*DAMPING,ALPHA=5000.,BETA=2.e-3
</PRE> <P> indicates that a damping matrix is created by multiplying the mass matrix with <SPAN CLASS="MATH"><B><IMG WIDTH="40" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" SRC="img2319.png" ALT="$ 5000.$"></B></SPAN> and adding it to the stiffness matrix multiplied by <!-- 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}$"></B></SPAN> <P> <PRE>
Example:
*DAMPING,STRUCTURAL=0.03
</PRE> <P> defines a structural damping value of 0.03 (3 <SPAN CLASS="MATH"><B><IMG WIDTH="17" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img1486.png" ALT="$ \%$"></B></SPAN>). This card must be part of a material description. <P> <P><P> <BR> Example files: beamimpdy1, beamimpdy2. <P> </body></html>