Modal Frequency Response Analysis

The modal method first performs a normal modes analysis to obtain the eigenvalues $λ_{i}$ and the corresponding eigenvectors $A$ of the system.

The response can be expressed as a scalar product of the eigenvectors $A$ and the modal responses, $d$ .

u = A d e^{i Ω t}

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

[- Ω^{2} A^{T} M A + A^{T} K A] d e^{i Ω t} = A^{T} f e^{i Ω t}

The modal mass matrix $A^{T} M A$ and the modal stiffness matrix $A^{T} K A$ are diagonal. If the eigenvectors are normalized with respect to the mass matrix, the modal mass matrix is the unity matrix and the modal stiffness matrix is a diagonal matrix holding the eigenvalues of the system. This way, the system equation is reduced to a set of uncoupled equations for the components of $d$ that can be solved easily.

The inclusion of damping, as discussed in the direct method, yields:

[A^{T} K A - Ω^{2} A^{T} M A + i G A^{T} K A + i A^{T} C_{GE} A + i Ω A^{T} C_{1} A] d e^{i Ω t} = X^{T} f e^{i Ω t}

Here, the matrices $A^{T} C_{G E} A$ and $A^{T} C_{1} A$ are generally non-diagonal. Then the coupled problem is similar to the system solved in the direct method, however of much lesser degree of freedom. It is solved using the direct method.

The evaluation of the equation of motion is much faster if the equations can be kept decoupled. This can be achieved if the damping is applied to each mode separately. This is done through a damping table TABDMP1 that lists damping values $g_{i}$ versus natural frequency $f_{i}^{f r e q}$ . If this approach is used, no structural element or viscous damping should be defined.

The decoupled equation is:

[- Ω^{2} m_{i} + i Ω c_{i} + k_{i}] d_{i} e^{i Ω t} = f_{i} e^{i Ω 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}^{f r e q})

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:

ζ_{i} = c_{i} / c_{c r} = g_{i} / 2

c_{c r} = 2 m_{i} ω_{i}

Q_{i} = 1 / 2 ζ_{i} = 1 / g_{i}

Modal damping is entered in to the complex stiffness matrix as structural damping if PARAM, KDAMP, -1 is used. Then the uncoupled equation becomes:

[- Ω^{2} m_{i} + (1 + i g (Ω)) k_{i}] d_{i} e^{i Ω t} = f_{i} e^{i Ω t}

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 Bulk Data Entry.

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 the 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. This is the default method since it is generally more accurate.
The applied load method generates a maximum of two residual vectors which are the dynamic load vector at a 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.

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.

Figure 1. Residual Vectors on the Result Accuracy of the Modal Frequency Response Analysis (FRA). Compared to the Accurate Direct Method

Modal Frequency Response Analysis with Enforced Motion

When a Modal Frequency Response analysis involves enforced motion (SPCD), there are 2 methods on how the solution is obtained.

Relative Method:
In this case, the solution proceeds in two stages. First, a static analysis with the enforced motion is solved to obtain the static displacements. Then, the dynamic analysis is solved using the previously calculated static displacements and the eigenvectors. This method is relatively less efficient, but leads to more accurate solutions and is the default method.
Total/Absolute Method:
In this case, the solution proceeds in a single stage and the calculation of static displacements is not needed. The contribution of modal dynamic load would directly come from applied displacement/velocity/acceleration at the SPCD degrees of freedom. This method is computationally efficient as it avoids calculation of static displacement vectors.

Refer to PARAM, ENFMETH to control the calculations with these methods.