Hello everyone!
From our earlier discussion on the Mode Superposition method,
we understood that it linearly combines the factored mode shapes from a modal analysis to calculate the structure's response.
However, there are specific cases where the mode-superposition method cannot be used,
such as systems with moderate or significant damping or systems with unsymmetric matrices
for example, those arising from vibroacoustic or bearing problems.
Hence, in such cases, performing Harmonic Analysis with the full method is required.
The full method doesn't use mode shapes from modal analysis,
as it solves the full system matrices directly.
This means that we don't have to worry about any numerical errors due to modal truncation,
but this also means that the full method is much more computationally expensive than the mode-superposition method.
Also, we can't cluster results since we don’t use any modal information.
While the mode-superposition method is recommended for most problems,
including lightly-damped systems, due to its speed and ability to cluster results,
this video lesson will discuss cases where it is recommended to use the Full Method for performing a Harmonic Analysis,
along with a demonstration.
So, let’s get started.
As we know, we need to perform a Harmonic Response Analysis
for understanding the steady-state dynamic response of the structure subjected to sinusoidally-varying loads.
The loads have known magnitude and frequency.
Also, multiple loads can be in-phase or out-of-phase with each other but are excited at the same frequency.
The results of a harmonic response analysis include displacements, stresses, and strains
that are a function of the excitation frequency and may be out-of-phase with the applied loads.
For any structure subjected to harmonic loading,
moderate-to-large damping values or unsymmetric matrices can generate a complex response.
Now, when we say moderate-to-large damping values,
it does not have to be critically damped or overdamped systems.
The Mode-superposition method assumes light damping and generally only uses ‘real’ mode shapes.
So, if the damping is relatively large, the mode shape can change and become complex modes – i.e., real and imaginary,
so the actual response with respect to the phase angle is a linear combination of the two.
To explain this further, in this example,
we have a spring with and without a damper in the middle of a cantilever beam with one end fixed, the other end excited.
For this first animation of spring with no damper, the mode shapes don't change.
We can see that the location of zero motion is stationary,
so going from phase angle 0 to 360 is just scaling the response.
For the second animation of spring with a damper, the stationary point travels along the beam.
And this cannot be represented with regular mode-superposition since it is not simply scaling a set of modes,
i.e., the shape changes with phase angle.
This behavior occurs with certain types of damping or unsymmetric matrices,
so that is why the full method is needed in such cases.
The Full Method of Harmonic Analysis is an exact solution
as compared to the approximations of the mode-superposition method.
However, the full method is computationally more expensive.
Using this harmonic analysis equation in the Full Method,
the matrix equation is solved directly in nodal coordinates using the Sparse matrix solver.
It is like a linear static analysis except that the complex numbers are included for the harmonic analysis.
When we study the results of harmonic analysis,
the solution points may be uniformly distributed across the frequency domain or non-uniformly distributed at user-defined locations.
The MSUP method is best suited for lightly-damped systems as discussed earlier.
But the full method does not have such restrictions on the degree of damping present in the model.
The Full Harmonic analysis can handle Rayleigh, Element, and Hysteretic damping
in the form of the Constant Structural Damping Coefficient.
And the viscoelastic damping can be defined under the Engineering Data by Prony Series.
There is an exception - the full method does not use modal information,
so modal damping cannot be specified with the full method.
Because the Full method does not rely on mode shapes and natural frequencies, the solution is exact,
which means that we don’t have to worry about modal truncation errors at each frequency.
However, the lack of modal information does mean that automatic clustering of results is not possible,
which can be important to capture peak responses accurately.
Note that the user can manually include additional frequencies for the solver to calculate,
such as near the natural frequencies of the structure, to ensure that critical frequencies are not missed when using the full method.
For this walkthrough session, let’s consider an example of Shock Absorbing Mounting Brackets or Shock Absorbing feet.
These feet are designed to absorb or dampen the vibrations generated by the objects that are mounted or connected to these feet.
The outdoor unit of a split Air Conditioner is a typical example where these feet are used to fix the unit to the ground.
For this demonstration, let’s consider one such Shock Absorbing foot.
We will use a fictitious rubber-like material that exhibits viscoelastic damping properties, to model this foot.
The use of viscoelastic damping cannot be included in the mode-superposition technique,
so we will go with performing a Harmonic Response analysis using the Full method.
So, let’s get started!
A Harmonic analysis system has already been added to the Workbench project page.
Let’s open Engineering Data.
For this demonstration, the fictitious material named “Rubber VE” has already been created with appropriate material properties.
Note that viscoelastic materials are hidden for harmonic response analyses by default,
so to view the viscoelastic properties, disable the “Filter Engineering Data” option.
Let’s close the Engineering Data and import the geometry.
Now open Mechanical for assigning materials.
Select the Damper foot body from the Geometry tree,
then go to the Details panel and assign material “Rubber VE”.
Let’s retain the default material assignment of Structural Steel for the remaining bodies.
Let’s retain the automatically generated bonded contacts and proceed with meshing.
Specify a body sizing of 5mm for the damper body and 3mm for the remaining 3 bodies.
Then generate the mesh.
Let’s define loads and boundary conditions.
Select the bottom face of the damper body and add Frictionless support.
Next, select the 2 cylindrical faces of the same body and specify a Cylindrical support with Tangential direction set as Free.
Now, let’s assume that the top face of the cylindrical body is connected to some external parts,
which would transfer loads leading to vibration to the damper foot.
To account for the external parts that are not modeled, we can make use of Remote Force.
Add a Remote Force load and scope it to the top face of the cylinder.
Specify a load of 50 N along the X-direction.
Assuming that the external parts are very stiff compared to the cylinder, set the behavior as Rigid.
Next, we need to constrain the rotation of the top face of this cylinder
which can be achieved by using the Remote Displacement boundary condition.
However, creating another remote boundary condition and scoping it to the same face will over constrain the model.
The correct method to do this is by promoting the remote point and then using the promoted remote point to define Remote Displacement.
Right-click on the Remote Force and select “Promote to Remote Point”.
A new Remote point gets created in the Model tree.
Also, notice that the scoping method of the Remote force has now changed to Remote Point
and the force is now scoped to the newly created remote point.
Add Remote Displacement boundary condition and scope it to the promoted remote point.
Set the rotations along the 3 axes to be zero to constrain the rotation of the top face.
Lastly, let’s specify Analysis Settings.
Set the minimum frequency as 100 Hz and the maximum frequency as 200 Hz.
Set Solution Intervals to 100 and select the Solution method as “Full”.
We are now set to start the solution.
It will take around 5 minutes to complete the solution.
We are now ready to analyze the results.
Add Deformation Frequency Response result
and scope it to the Remote point.
Select the Orientation along the X-axis which is our direction of loading.
Now evaluate the result.
You can see the amplitude and phase angle plots with respect to frequency.
If you go to the Details panel of Frequency response and expand the Results section,
you can get to know the maximum amplitude and the frequency at which it occurs along with its corresponding phase angle.
Now let’s create a contour plot for this frequency response.
The easiest way to do this is by right-clicking on Frequency Response and selecting “Create Contour Result”.
Notice that the frequency and sweeping phase angle values have been automatically filled
and they match with the values from the Results section of the Frequency response chart.
The only difference here is the sign of the Phase Angle.
The Phase Angle of the contour result has the same magnitude as the frequency result type but an opposite sign.
The sign is reversed so that the response amplitude of the frequency response plot for that frequency and phase angle
matches with the contour results.
Next, let’s evaluate Equivalent stress on the damper body for the same frequency and phase angle.
Finally, let’s evaluate Deformation Phase Response to see how much the frequency response lags behind the applied load
over a phase period.
Scope it to a Remote point and evaluate it for the same frequency.
This completes the demonstration of Harmonic analysis using the Full Method.
So let’s summarize key learning points.
The mode Superposition method linearly combines the factored mode shapes from a modal analysis
to calculate the structure's response.
It is generally recommended for most problems due to its speed and ability to cluster results.
It is an ideal option for lightly damped systems.
The full method solves the full system matrices directly.
It is required when the mode superposition method cannot be used,
such as systems with appreciable damping or systems with unsymmetric matrices.
I hope you have found this video informative.
Don’t forget to check out our other courses to discover more useful learning resources.