Mode displacement method

You can use the mode displacement method to recover dynamic physical responses from modal responses. It assumes the physical responses are the linear combination of modal contributions from normal modes.

Using the mode displacement method, motion responses are recovered by:

\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}=[{{\Phi }{n}}]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\gamma }+[{{\Phi }{s}}]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}_{s}}	Equation 1

\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{x}}=[{{\Phi }{n}}]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{\gamma }}+[{{\Phi }{s}}]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{x}}}_{s}}	Equation 2

\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{x}}=[{{\Phi }{n}}]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{\gamma }}+[{{\Phi }{s}}]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{x}}}_{s}}	Equation 3

where

[\Phi_n]	is the matrix of normal modes;
[\Phi_s]	is the matrix of constraint modes;
\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{x}},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{x}}	are displacement, velocity, and acceleration;
\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\gamma },\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{\gamma }},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{\gamma }}	are vectors of modal displacement, velocity, and acceleration;
{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}_{s}}	is vector of enforced displacement.

Similarly, Response Dynamics recovers dynamic stresses using the following equation:

\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\sigma }=[\Phi _{n}^{\sigma }]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\gamma }+[\Phi {s}^{\sigma }]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}{s}}	Equation 4

where

\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\sigma }	is stress;
[\Phi_n^\sigma]	is matrix of modal stresses;
[\Phi_s^\sigma]	is matrix of modal stresses of constraint modes.

Equation 4 can also be used to calculate other responses by simply substituting the stresses with strains, element forces, shell stress resultants, or reaction forces.

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Mode acceleration method

Modal response calculation for transient analysis

Modal response calculation for frequency analysis

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Response spectrum analysis

Modal damping

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Shell stress resultants

Theory of Fast RMS modal response computing

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Mode displacement method, Simcenter 3D 2021.1 Series

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