Modal response calculation for transient analysis

For transient analysis, the modal responses

\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\gamma }(t),\text{ }\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{\gamma }}(t),\text{ }\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\dot{\gamma }}(t)

are determined by time integration.

Using modal coordinates, the equation of motion can be written in modal space as:

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

Equation 1

where

\left[ \begin{matrix} \ddots & {} & {} \ {} & m & {} \ {} & {} & \ddots \ \end{matrix} \right],[\left[ \begin{matrix} \ddots & {} & {} \ {} & c & {} \ {} & {} & \ddots \ \end{matrix} \right],\text{and }[\left[ \begin{matrix} \ddots & {} & {} \ {} & k & {} \ {} & {} & \ddots \ \end{matrix} \right]	are diagonal modal mass, damping, and stiffness matrices;
t	is time;
{\left[ \text{ } \right]}^{T}	is operation of transposition of a matrix;
[\Phi_n]	is matrix of normal modes;
[\Phi_m]	is matrix of effective mass of enforced motion DOF;
[\Phi_c]	is matrix of effective damping of enforced motion DOF;
f_i	are applied forces;
s	is index that denotes restrained DOF.

In Response Dynamics, Equation 1 is integrated using the convolution integral (or Duhamel's integral).

Note:

The third (damping) term in the right hand side of Equation 1 is not included in Response Dynamics.

Initial conditions

For transient analysis in Response Dynamics, you can set the initial condition to zero, or you can define a Static Deformation initial condition. If the initial condition is zero, you assume the model is initially undeformed, which means

{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\gamma }}_{0}}=0

where

{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\gamma }}_{0}}	are the initial modal displacements

If you want the software to determine the initial configuration using the load at the initial time point, you can select Static Deformation as the Initial Condition. The software then determines the initial deformation by solving:

[k]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}{0}}={{\widehat{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{f}}}}^{0}}={{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{f}}^{0}}-\left( \left[ {{M}{is}} \right]+\left[ {{M}{ii}} \right]\left[ {{\Phi }{s}} \right] \right)\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{x}}_{s}^{0}

Equation 2

where

[k]	is the stiffness matrix;
{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x}}_{0}}	= the initial displacement
{{\widehat{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{f}}}}^{0}}	is the effective external forces (including enforced motion excitations)
i	is unrestrained DOF
\left[ {{\Phi }_{s}} \right]	is matrix of constraint modes of enforced motion DOF

Equation 2 can be written in the modal space as

\left[ \begin{matrix} \ddots & {} & {} \ {} & k & {} \ {} & {} & \ddots \ \end{matrix} \right]{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\gamma }}{0}}={{\left[ {{\Phi }{n}} \right]}^{T}}{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\hat{f}}}^{0}}

or

| {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\gamma }}{0}}={{\left[ \begin{matrix} \ddots & {} & {} \ {} & k & {} \ {} & {} & \ddots \ \end{matrix} \right]}^{-1}}{{\left[ {{\Phi }{n}} \right]}^{T}}\left( {{\left. {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{f}}}{i}} \right|}{t=0}}-\left( \left[ {{M}{is}} \right]+\left[ {{M}{ii}} \right]\left[ {{\Phi }{s}} \right] \right){{\left. {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{x}}}}{s}} \right|}_{t=0}} \right) | Equation 3 | | --- | --- |

Initial velocities

By default, the initial velocities of the model are assumed to be zero. Or, in the modal space, the modal velocities are zeros.

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