Modal reduction

Response Dynamics uses modal approaches to perform transient, frequency, random, and response spectrum analyses. Before performing a Response Dynamics, you solve the response dynamics solution to reduce your physical model to a modal model.

This topic discusses the modal reduction as:

• Normal modes and dynamic equations of motion

• Constraint modes and enforced motion excitations

Normal modes and dynamic equations of motion

Equations of motion can be written in terms of physical DOF (for example, nodal DOF for a finite element model):

\left[ \begin{matrix} {{M}{ii}} & {{M}{is}} \ {{M}{si}} & {{M}{ss}} \ \end{matrix} \right]\left{ \begin{matrix} {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{x}}}}{i}} \ {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{x}}}}{s}} \ \end{matrix} \right}+\left[ \begin{matrix} {{C}{ii}} & {{C}{is}} \ {{C}{si}} & {{C}{ss}} \ \end{matrix} \right]\left{ \begin{matrix} {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{x}}}}{i}} \ {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{x}}}}{s}} \ \end{matrix} \right}+\left[ \begin{matrix} {{K}{ii}} & {{K}{is}} \ {{K}{si}} & {{K}{ss}} \ \end{matrix} \right]\left{ \begin{matrix} {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}}{i}} \ {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}}{s}} \ \end{matrix} \right}=\left{ \begin{matrix} {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{f}}}{i}} \ {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{R}}}{s}} \ \end{matrix} \right}

Equation 28-1.

where

subscript i denotes unconstrained DOF;

subscript s denotes constrained DOF;

{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{f}}_{i}}	is vector of forces associated with unconstrained DOF;
{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{R}}_{i}}	is vector of reaction forces of constraints.

M, C , K are mass, damping, and stiffness matrices, respectively.

Let {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}_{i}} be represented by a set of normal modes:

{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}{i}}=\left[ {{\Phi }{n}} \right]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\gamma }

Equation 28-2.

where

[Φn]	is the modal matrix;
\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\gamma }	is the vector of modal coefficients.

Substitution of Equation 2 into Equation 1 produces the following differential equations for the modal coefficients:

\left[ \begin{matrix} \ddots & {} & {} \ {} & m & {} \ {} & {} & \ddots \ \end{matrix} \right]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{\gamma }}+\left[ \begin{matrix} \ddots & {} & {} \ {} & c & {} \ {} & {} & \ddots \ \end{matrix} \right]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{\gamma }}+\left[ \begin{matrix} \ddots & {} & {} \ {} & k & {} \ {} & {} & \ddots \ \end{matrix} \right]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\gamma }={{\left[ {{\Phi }{n}} \right]}^{T}}{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{f}}{i}}

Equation 28-3.

where

\left[ {\begin{array}{{20}{c}} \ddots &{}&{} \ {}&m&{} \ {}&{}& \ddots
\end{array}} \right],,,,\left[ {\begin{array}{{20}{c}} \ddots &{}&{} \ {}&c&{} \ {}&{}& \ddots
\end{array}} \right],,,,\left[ {\begin{array}{*{20}{c}} \ddots &{}&{} \ {}&k&{} \ {}&{}& \ddots
\end{array}} \right],,

are diagonal modal mass, damping, and stiffness matrices respectively;

the operator [ ]T denotes transposition of matrix.

Constraint modes and enforced motion excitations

If an enforced motion (or base excitation) is applied, that is if the restrained DOF in Equation 1 are defined as motion, then the sought DOF can be represented as

{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}{i}}=\left[ {{\Phi }{s}} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}{s}}+\left[ {{\Phi }{n}} \right]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\gamma }

Equation 28-4.

where [Φs]=-[Kii]-1[Kis] is the matrix of constraint modes.

Substitution of Equation 4 into the upper part of Equation 1, produces the following differential equations for the modal coefficients:

\left[ \begin{matrix} \ddots & {} & {} \ {} & m & {} \ {} & {} & \ddots \ \end{matrix} \right]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{\gamma }}+\left[ \begin{matrix} \ddots & {} & {} \ {} & c & {} \ {} & {} & \ddots \ \end{matrix} \right]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{\gamma }}+\left[ \begin{matrix} \ddots & {} & {} \ {} & k & {} \ {} & {} & \ddots \ \end{matrix} \right]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\gamma }={{\left[ {{\Phi }{n}} \right]}^{T}}\left( \underset{{}}{\mathop{{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{f}}}{i}}-\left[ {{\Phi }{m}} \right]{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{x}}}}{s}}-\left[ {{\Phi }{c}} \right]{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{x}}}}{s}}}}, \right)

Equation 28-5.

where

[Φm]=[Mii ][Φs]+[Mis] is matrix of effective mass of DOF with enforced motion;

[Φc]=[Cii][Φs]+[Cs] is matrix of effective damping of DOF with enforced motion (not included in Response Dynamics).

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