Hello everyone,
In response spectrum analyses, we select a mode combination method to estimate the peak response of our structure.
However, are these mode combination methods applicable for a wide frequency range?
Also, what if the number of modes we have is sufficient to cover the frequency range but does not have sufficient effective mass?
In this video, we’ll answer these questions and investigate techniques to obtain more accurate response spectrum results.
Ready?
Let’s get started.
Recall that, in response spectrum analyses, the modal responses are combined,
and the interaction of modes and the effect of damping may also be included.
Note that the frequency content of our system can span a wide range,
so can we treat all modes the same way with mode combination techniques?
To better answer this question, let’s look at a simple system of two cubes connected to a base, each with different springs.
As we excite the base vertically, the two cubes will vibrate, and we see differences in the response based on the excitation frequency.
Here, the left animation shows a response with a low frequency excitation on the base and the right animation shows the response with a high frequency excitation.
Can you see the difference here?
On the left side, when one cube is going up, the other one is going down.
If we plot the two deformations along time, we can see that there’s a phase difference between the two waves.
Now let’s look at the right-side animation where higher-frequency excitation is applied;
here, although the amplitudes of the two cubes are different, they reach their peak displacements in the same direction at the same time.
We say that the left animation shows out-of-phase responses, which may also be referred to as “periodic” response,
while the right one shows in-phase responses, which may demonstrate a “rigid” response.
In a response spectrum analysis,
we would combine periodic responses with the mode combination methods since the peak responses often don’t occur at the same time.
However, for rigid response, we’d prefer to add them algebraically since they may be in-phase.
Our next question may be, “How do we determine if the response is periodic or rigid?"
This is best addressed by looking at an input spectrum, where we generally separate the response into three regions.
The rigid response region is the high frequency plateau range of the curve.
The starting point of the plateau is important. We define the frequency at this point as f_ZPA,
and the spectral value at this point ZPA, or zero period acceleration.
The periodic region is located in the low-frequency region.
The end of the periodic region is generally characterized by the peak of the input spectrum.
We call the frequency at peak response f_sp, where SP stands for spectral peak.
Now, we have the rigid response and period region highlighted, the in-between region is the transition area from period to rigid response.
Between f_SP and f_ZPA, modes may be thought to have both periodic and rigid components.
The next step is to determine how to incorporate the rigid response calculations in our mode combination method for response spectrum analyses.
We noted earlier that mode combination methods like SRSS, CQC, and Rosenblueth are suitable for the periodic region,
and the rigid response should be added algebraically.
How do we account for the transition region?
In this region, response of an individual mode is decomposed into two parts, the periodic part R_p, and the Rigid part R_r,
controlled by one coefficient alpha, that is a number between 0 and 1.
We can see that if alpha is 0, the modal response becomes purely periodic and if alpha is 1, it’s purely rigid.
For mode combination, the periodic response and rigid response are combined separately.
We find the portion of rigid response of all the modes and combine them by plain summation.
And for the periodic response of all the modes in the transition region, they are combined by SRSS, CQC or Rosenblueth methods.
The periodic and rigid terms are then combined via SRSS.
We see that the coefficient alpha plays an important role, as it defines the portion of rigid response.
Ansys Mechanical provides two popular methods to define alpha:
the Lindley-Yow method and Gupta method.
For the Lindley-Yow method, the coefficient alpha is defined based on the zero period acceleration of the input spectrum and the spectral value of each mode.
We can turn the input spectrum curve 90 degrees to understand the Lindley-Yow method better.
Alpha is 0 when the spectral value is smaller than ZPA.
Alpha is 1 when the spectral value equals ZPA and starts to decrease for increasing spectral values.
For the Gupta method, the equations are a little bit more involved.
But it’s actually not too hard to understand the philosophy here.
Basically, the Gupta method defines alpha based on the frequency, the horizonal axis, instead of spectral value.
It uses a low frequency f1 and a high frequency f2 to determine the cutoff of periodic response and rigid response.
Below f1, alpha is 0 and above f2, alpha becomes unity.
Everything in-between is defined by this equation.
And as we see here, f1 is just the frequency at peak of the curve and f2 is defined based on f1 and fzpa.
Basically, the key difference between the two methods is that the Lindley-Yow method uses spectral values, or the y-axis, to define alpha,
while the Gupta method uses frequencies, or the x-axis, to define alpha.
In addition to a separate treatment of rigid responses,
we also have another feature, called Missing Mass Response, to improve the accuracy of our response spectrum calculations.
In a modal analysis, we do not extract all the modes of a structure since it is computationally expensive and generally unnecessary.
A way to identify if enough modes are extracted for a structure is to check the ratio of effective mass to total mass.
However, what should we do if we have extracted more than enough modes to cover our frequency range of interest,
but the ratio of effective mass to total mass is relatively low?
The basic idea of the missing mass method is that we can calculate how much mass is missing from the modal analysis and account for this quantity in the response spectrum analysis.
Note that beyond f_zpa, the rigid response dominates for these high frequency modes.
Thus, we know that the ‘missing mass’ are frequencies much higher than f_zpa acting in-phase.
We also know that if we solved for all of the modes, the effective mass would equal the actual mass.
Therefore, if we apply an acceleration field of ZPA in a static analysis and subtract our modal inertia forces,
we are left with the “missing” inertia force from the high-frequency modes.
To explain this method visually, from this graph,
what we did first is to assume ZPA as the spectral value for all existing modes including the neglected ones.
Here we use the area under the ZPA line to visually represent the total inertia force, although mathematically, inertia force is not exactly the area.
Then we find the summation of the inertia force for all known modes, marked by green here.
Subtract the green area from the yellow area, the remaining area becomes the inertia force for the missing mass.
Now with the missing inertia force, and with the stiffness matrix of the structure, we can easily calculate the missing mass deformation response.
Lastly, the missing mass response is combined with the previously calculated response by SRSS method.
Where the first term under the square root is the generalized form of mode combination method while the second term accounts for the response calculated by the missing mass.
It’s worth noting that the rigid response and the missing mass effect can be used together, if needed.
For such cases, the total response equation term has another term under the square root sign accounting for the rigid response.
Now let’s go to workshop models to check how the Rigid response and Missing mass features are specified, and how this affects the results.
Here we are looking at a modal analysis of a generic tower structure with all the bottom points fixed on the ground.
From the modal analysis, we can see that 100 modes are extracted for the structure.
Let’s go back to Workbench schematic, drag a Response Spectrum Analysis from the left toolbox and drag it over the engineering data,
Geometry, Model, Setup and Solution cells of the Modal analysis, and release the mouse.
Then let’s drag another response spectrum analysis over the modal analysis and rename it to "Response Spectrum rigid response or missing mass ".
Now double click on the setup of the first response spectrum analysis to open Ansys Mechanical.
Let’s insert a Response Spectrum Acceleration here.
Choose All Supports for boundary condition. Set direction to Z, and fill the table with the given input spectrum here.
We will do the same thing for the second response spectrum analysis.
But, this time, we will turn on the Rigid Response Effect here.
For the type, we choose the Gupta method.
Looking at the input spectrum curve, we see that the peak spectral value is at 5 Hz, which is used as f 1, defining the beginning of the rigid response.
And the plateau range for the high frequencies starts from 10 Hz, which is identified as f_zpa.
Based on the equation on Gupta Method, we can find f2
Note that, after f 2 it is assumed to be pure rigid response.
Now, let’s run these two response analyses to compare the results.
We can see here with rigid response on, the total deformation and the deformation in the excitation direction, z direction, become larger.
This means the plain summation of the rigid response frequencies and part of the transition region frequencies leads to a larger total response compared to SRSS.
Such change is expected when rigid response frequencies exist for the structure.
In this demonstration, the structure and the input spectrum are tuned to show the differences in results when neglecting and including rigid responses.
In practical use, if there’s not much rigid response in the defined region,
there may not be any noticeable difference when including rigid responses.
On the other hand, if the rigid response is incorrectly specified on low frequency regions where the periodic response is dominant,
this might lead to incorrect results, so one must exercise caution when defining the rigid response parameters.
Now let’s use another input spectrum for this model and demonstrate the missing mass feature.
The new input spectrum covers 0.4 Hz to 6 Hz.
In this case, to cover the input spectrum range, we actually only need 50 modes.
Let’s say we set 50 modes to be extracted and run the modal analysis again.
As you can see here, the highest frequency is about 8 Hz, larger than the input spectrum.
However, if we go to 'Solution Information' and check the 'Participation factor summary',
we can see that the ratio of effect mass to total mass is low.
For example, in the excitation direction, z direction, only 67% mass is extracted, which means 33% mass is missing.
In this simple model, extract more modes is not a problem, but in some practical engineering problems,
to increase the effective mass , thousands or more modes will be required, which is not efficient.
And this is when missing mass method can be useful.
Now let’s see if turning on missing mass feature effect the response of this structure.
Let’s go the second response spectrum analysis, change the input spectrum to the new values.
Now turn off the rigid response and turn on the missing mass effect.
For ZPA, we will input the spectral value of the plateau in the curve.
Now, let’s run the two analyses, one without missing mass effect and the other with missing mass effect, to compare the results.
Here we can see that with missing mass effect, the deformation results are slightly larger, which is expected.
Note that in this demonstration, the model and the input spectrum is specially crafted to show the effect of missing mass.
This concludes the demo.
Now, let’s summarize what we learned in the video.
Generally speaking, mode combination methods such as SRSS, CQC, and Rosenblueth are intended for periodic response,
where modal responses are out-of-phase with each other.
The rigid response effect can algebraically sum modes that are in the high frequency range instead of using mentioned mode combination methods.
In the transition region, either the Lindley-Yow or Gupta methods define a fraction of the modal response that is rigid.
Although extracted modes can cover the frequency range of interest, they may not have enough effective mass,
so the Missing Mass Method provides a technique to account for the inertia forces of the mass not included in the modes.
Either or both rigid responses and missing mass responses can be included in a response spectrum analysis.
I hope you found this video informative.
Thank you for watching and do check out courses.ansys.com for more useful learning resources.