Linear Modal Transient Response

The state vector $u$ can be expressed as a scalar product of the eigenvectors $A$ and the modal responses $v$ .

$u = A v$

The equation of motion without damping is then transformed into modal coordinates using the eigenvectors:

$A^{T} M A \ddot{v} + A^{T} K A v = A^{T} f$

The modal mass matrix $A^{T} M A$ and the modal stiffness matrix $A^{T} K A$ are diagonal. This way the system equation is reduced to a set of uncoupled equations for the components of $v$ that can be solved easily.

The inclusion of damping yields:

$A^{T} M A \ddot{v} + A^{T} C A \dot{v} + A^{T} K A v = A^{T} f$

Here, the matrices $A^{T} C A$ are generally non-diagonal. Then coupled problem is similar to the system solved in the direct method, but of a much lesser degree of freedom. The solution of the reduced equation of motion is performed using the Newmark method.

The decoupling of the equations can be maintained, if the damping is applied to each mode separately. This is done through the damping tables, that can be defined using these methods:

TABDMP1 lists damping values $g_{i}$ versus natural frequency $f_{i}$ .
TABDMP2 lists damping values $g_{i}$ versus range of mode indices.

The decoupled equation is:

$m_{i} {\ddot{v}}_{i} (t) + c_{i} {\dot{v}}_{i} (t) + k_{i} v_{i} (t) = f_{i} (t)$

or

${\ddot{v}}_{i} (t) + 2 ζ_{i} ω_{i} {\dot{v}}_{i} (t) + ω_{i}^{2} v_{i} (t) = \frac{1}{m_{i}} f_{i} (t)$

Where,

$ζ_{i} = c_{i} / (2 m_{i} ω_{i})$: Modal damping ratio
$ω_{i}^{2}$: Modal eigenvalue

Three types of modal damping values

$g_{i} (f_{i})$ can be defined:

$G$: Structural damping
$C R I T$: Critical damping
$Q$: Quality factor

They are related through the following three equations at resonance:

$G = ζ_{i} = \frac{c_{i}}{c_{c r}} = \frac{g_{i}}{2}$

$C R I T = c_{c r} = 2 m_{i} ω_{i}$

$Q = Q_{i} = \frac{1}{2 ζ_{i}} = \frac{1}{g_{i}}$

Residual Vector Generation (Increases Accuracy)

The accuracy of the modal method can be vastly improved by adding the displacement vectors of a static analysis based on the dynamic loading to the matrix of eigenvectors $X$ . These vectors are frequently referred to as residual vectors, the method as modal acceleration.

There are two ways this is implemented.

The unit load method generates residual vectors based on static loads, which are unit vectors at the dynamic load degrees of freedom. That is, the static loads for the residual vector generation are unit vectors at the degrees of freedom, where the dynamic load is applied. The number of residual vectors is equal to the number of loaded degrees of freedom.
The applied load method generates a maximum of two residual vectors which are the dynamic load vector at loading frequency of zero. If the real and the imaginary parts of the dynamic load are the same, or if one of them is zero, only one of them is used. This is the default method since it is generally more efficient.

In the case of excited displacements, the residual vectors are obtained by solving static load cases with unit displacements at the same degrees of freedom as the dynamic excited displacement degrees of freedom.

Run Linear Modal Transient Analysis

Loads and Boundary Conditions are defined in the Bulk Data Entry section of the input deck. They need to be referenced in the Subcase Information Entry section using an SPC statement and a DLOAD statement in a SUBCASE. Similar to mechanical loads, temperature loads (TEMP/TEMPD/TEMPADD) can be applied to Linear Transient Analysis via TLOAD1/TLOAD2 Bulk Data Entries referenced on a DLOAD Subcase Information Entry.

Residual vectors can be activated using the Subcase statement RESVEC with the options APPLOD or UNITLOD. They are computed by default. Residual vectors are always generated if enforced displacements, velocities or accelerations are defined. Residual vectors are also calculated for viscous damping DOF. These are created by default and can be turned off with the RESVEC option NODAMP. In addition, if there is USET U6 data, residual vectors will be calculated if the AMSES or AMLS eigensolver is used. USET U6 residual vectors will not be calculated if the Lanczos eigensolver is used.

When residual vectors are included, Inertia Relief can be applied to unconstrained models. A SUPORT1 Subcase Information Entry references the boundary conditions that restrain the rigid body motions. These restraints can also be defined without subcase reference using the SUPORT Bulk Data Entry or automated using PARAM,INREL, -2.

Only one transient subcase can be defined. Initial conditions cannot be defined if the modal method is used. A METHOD statement is required for the modal method to control the normal modes analysis. The METHOD statement can refer to either EIGRL or EIGRA data.

The analysis time step and termination time need to be defined through a TSTEP(TIME) subcase reference. In order to save computational effort, previously saved eigenvectors can be retrieved using the EIGVRETRIEVE subcase statement.

In addition to the various damping elements and material damping, uniform structural damping $G$ is applied using PARAM, G.

Modal damping can be applied using the SDAMPING reference of a damping table TABDMP1.