(SM_Comp_FRF)=
# Comparing Sierra SM Explicit Transient to Direct and Modal FRF

FRFs are a matrix of relationships from forced input to either
displacement, velocity, or acceleration output. Typically, system
response is accessed using acceleration.

**Frequency Response Functions.** The transfer function $[H]$ relates
the force input to the displacement between two points in the system.
The transfer function is symmetric and is formed as a function of mass,
damping, and stiffness. The transfer function is differentiable and the
relationship of the force to the acceleration is shown using the
following in matrix form:

$$
\bar{A} = \ddot{[H]} \bar{F}
$$

More information can be found in the theory manual.

**Mesh.** Figure {numref}`%s <fig:TF_setup>` shows a sample mesh that
was used both as an input for Sierra/SD Modal and Direct FRF as well as
for the Sierra Solid Mechanics Code - Adagio. The Node where force is
applied is connected to the beam using a network of rigid Rbars and the
force is applied in the Z-direction.

:::{figure-md} fig:TF_setup

![](TF.png)

[Cantilever Beam FRF example problem](/tests/Salinas_rtest/training/exampleproblem/frf/direct/smFRFcomp/beam_frf.e)
:::

Three inputs are considered: [modalFRF](/tests/Salinas_rtest/training/exampleproblem/frf/direct/smFRFcomp/frf_m.inp), [directFRF](/tests/Salinas_rtest/training/exampleproblem/frf/direct/smFRFcomp/frf_dir.inp), and [Sierra/SM explicit transient](/tests/Salinas_rtest/training/exampleproblem/frf/direct/smFRFcomp/beam_sm.i).

The input below shows the relevant portions of the direct FRF input file.
The keyword `alpha = 5` sets the mass damping of the system. The
`frequency` section has the frequency range from .1-50Hz at .1Hz
increments. A general rule of thumb is that the highest frequency mode
requested in the `solution` section should be at least 1.5x the max
frequency in the `frequency` section.

:::{input}
LOADS
 nodeset 500
 force = 0 0 1
 scale = 1
 function = 1
END

FUNCTION 1
  type LINEAR
  name "white noise"
  data 0.0 1.0
  data 200. 1.0
END

DAMPING
 alpha = 5
END

FREQUENCY
  freq_min = .1
  freq_step = .1
  freq_max = 50
  acceleration
  disp
  nodeset 2
END
:::
:::{figure-md} fig:SM_compare

![](sm_compare.png)

FRF Z-axis Modes ResultsAcceleration of end node in the Z-axis direction.
:::

Figure {numref}`%s <fig:SM_compare>` compares modal and direct FRF with
the same damping. There are enough modes for the modal FRF to show
nearly exact agreement to the direct frf results. Each of the
frequencies used for the adagio input show reasonable agreement. The
discrepancies seen are possibly due to the possibility that the alpha
damping in adagio is not one-to-one related to the alpha damping in
Sierra/SD.

Table {numref}`%s <tbl:FRFTimes>` shows each method's run time. A caveat
should be noted here that 10 cycles were used in the Adagio input to
ensure that the system reached steady state. Reducing the number of
cycles reduces the run time proportionally. In addition, with complex
systems, eigen solution run time added to the modal FRF solver time may
approach the direct FRF solution time. It also should be noted that
Adagio run was performed with the knowledge of mode frequency locations.
If it were not, it is possible that the frequencies needed to plot would
be closer together and more numerous.

:::{table} Run Times (min:sec).
:name: tbl:FRFTimes

| Method                  | Time   |
|:------------------------|:-------|
| Modal FRF (20 modes)    | 00:09  |
| Direct FRF              | 02:41  |
| Sierra SM (8 frequencies) | 129:48 |
:::
