Mode acceleration method

For transient and frequency events, you can use the mode acceleration method to recover dynamic physical responses from dynamic modal responses. It assumes the displacements are calculated by adding the acceleration effects (or the static results due to the inertia forces) to the static responses.

The dynamic equations of motion of a finite element model can be written as

\left[ {{M}{ii}} \right]{{\ddot{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}}{i}}+[{{C}{ii}}]{{\dot{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}}{i}}+\left[ {{K}{ii}} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}{i}}={{\hat{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{f}}}_{i}}	Equation 1

where

\left[ {{M_{ii}}} \right],,,[{C_{ii}}],,,\left[ {{K_{ii}}} \right]	are the mass, damping, and stiffness matrices associated with the unrestrained DOF
{{x}{i}},{{\dot{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}}{i}},{{\ddot{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}}_{i}}	are the displacement, velocity, and acceleration vectors

and

{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\hat{f}}}{i}}={{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{f}}{i}}-\left[ {{M}{is}} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{x}}}{s}}-\left[ {{C}{is}} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{x}}}{s}}-\left[ {{K}{is}} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}{i}}

are the effective external forces

in which

{{\hat{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{f}}}_{i}}	are the applied forces
{{\dot{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}}{s}},{{\ddot{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}}{s}}	are the enforced velocity and acceleration defined at the restrained DOF
\left[ {{M_{is}}} \right],,,[{C_{is}}],,,\left[ {{K_{is}}} \right]	are the coupled mass, damping, and stiffness matrices between restrained DOF and unrestrained DOF

The displacement can be calculated from the velocity and acceleration by rewriting Equation 1 into the following format:

\left[ {{K}{ii}} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}{i}}={{\widehat{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{f}}}}{i}}-\left[ {{M}{ii}} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{x}}}{i}}+[{{C}{ii}}]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{x}}}_{i}}	Equation 2

Velocities and accelerations are still recovered directly from normal modes responses as:

{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{x}}}{i}}=\left[ {{\Phi }{n}} \right]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{\gamma }}+\left[ {{\Phi }{s}} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{x}}}{s}}	Equation 3
{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{x}}}{i}}=\left[ {{\Phi }{n}} \right]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{\gamma }}+\left[ {{\Phi }{s}} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{x}}}{s}}	Equation 4

where

\Phi_n	is matrix of normal modes.

Substituting Equations 3 and 4 into Equation 2, displacements can be calculated by:

{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}{i}}=\left[ {{\Phi }{a}} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{f}}{i}}+\left[ {{\Phi }{aa}} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{x}}}{s}}+\left[ {{\Phi }{a\nu }} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{x}}}{s}}+\left[ {{\Phi }{s}} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}{s}}+\left[ {{{\ddot{\Phi }}}{n}} \right]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{\gamma }}+\left[ {{{\dot{\Phi }}}_{n}} \right]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{\gamma }}

Equation 5

where

\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;
[\Phi_a]	is matrix of attachment modes due to unit applied force;
[{{\Phi }_{aa}}]	is matrix of equivalent attachment modes due to enforced acceleration;
	=-{{[{{K}{ii}}]}^{-1}}\left( [{{M}{ii}}][{{\Phi }{s}}]+[{{M}{is}}] \right)
\left[\Phi_{a\nu}\right]	is matrix of equivalent attachment modes due to enforced velocity;
	=-\left[K_{ii}\right]^{-1} \left([C_{ii}][\Phi_s]+[C_{is}] \right) Note: This term is not included in Response Dynamics.
[\Phi_s]	is matrix of constraint modes of enforced motion DOF;
	=-[K_{ii}]^{-1}[K_{is}]
[\ddot{\Phi}_n]	=-\frac{1}{\omega_n^2}{ \phi_n }
[\dot{\Phi}_n]	=-\frac{2\xi_n}{\omega_n}{\phi_n}

and where

\omega_n	is natural frequency,
\xi_n	is damping ratio (factor),
{\phi_n}	is mode shape of the n-th eigenmode.

Equation 5 can also be used to calculate dynamic stresses, strains, element forces, shell resultants, and reaction forces by simply using corresponding results instead of displacement shapes in the equation.

Rigid body motion

If rigid body modes exist, the responses can be calculated by including the rigid body motion as:

{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}{i}}=\left[ \Phi {n}^{R} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\gamma }}{R}}+\left[ {{\Phi }{a}} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{f}}{i}}+\left[ {{\Phi }{aa}} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{x}}}{s}}+\left[ {{\Phi }{a\nu }} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{x}}}{s}}+\left[ {{{\ddot{\Phi }}}{n}} \right]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{\gamma }}+\left[ {{{\dot{\Phi }}}{n}} \right]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{\gamma }}+\left[ {{\Phi }{s}} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}_{s}}

Equation 6

where

\left[ \Phi_n^R \right]	is the matrix of rigid body modes;
{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\gamma }}_{R}}	is the modal displacement of rigid body modes.

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