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Consider a small piece cut out of a structure that has non-zero internal stress.
To maintain equilibrium, there must be some forces applied to the boundary of the piece. (Of course when it was part of the complete structure, those forces came from the stress in the adjacent parts of the structure.)
When you deform the piece further, those external forces ...
5
They way to answer this question is transform the dynamics model into modal coordinates, and see what happens to the force term.
Suppose we can describe the stiffness and mass properties of the structure as matrices $\mathbf K $ and $\mathbf M$, and its displacement as a vector $\mathbf x$, in a physical coordinate system, and we apply a vector of forces ...
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Take the Fourier Transform of the time varying driving force, this will give the frequency content of the driving force. Multiple modes of vibration can be driven at once, and will superpose with each other, but time varying driving forces with a frequency content that are high at frequencies near a particular resonant frequency will mainly drive the ...
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The frequency and ground acceleration and other pertinent data are recorded in a seismic design espectra which is unique for each site and varies with the type of the soil on the side.
Usually building department will have that information.
For example the following is the link to Los Angeles building department site for design espectra.
https://data2.scec....
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The only way you can get negative eigenvalues is by including stress stiffness effects (sometimes called "geometric stiffness"), if there are compressive stresses which would cause the plate to buckle.
There can be zero eigenvalues if the plate can move as a rigid body, of course, and they might be calculated as small negative numbers, but those should be a ...
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Firstly, applying a constant load will not excite any modes. You need a time-varying load to induce vibration (e.g. wind, seismic, blast). Which modes are excited will depend on the frequency content of the applied loading.
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As modal analysis is performed in the frequency domain, your modes and amplitudes are also in the frequency domain.The modal amplitudes ($q_i$) are scaled by the same amount as the applied load. If you want to know the displacement at each mode, then just multiply $q_i$ by whatever the magnitude of the load (for the mode) is scaled by. The maximum amplitude ...
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The body of your question (the title is somewhat different) asks about computing the frequency response function (FRF) between a multi-input, multi-output (MIMO) system. First off, the numerical transfer function/FRF between two time domain signals is usually calculated by calculating the cross power spectrum between the two signals and dividing it by the ...
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Disclaimer: Base on the question, I am suspecting you are looking for a theoretical explanation. Unfortunately I am not in a position offer any insight in short order.
But I can offer you following two ideas and source code to access data from three axis ADXL345 accelerometer to help you with your endeavors.
Three axis ADXL345 accelerometer implementation ...
answered Sep 10 '15 at 12:28
Mahendra Gunawardena
6,5435 gold badges22 silver badges59 bronze badges
2
The most basic purpose of the measurements is system identification - i.e. to determine the modal frequencies, mode shapes, and damping factors for the modes in some frequency range.
You can use that information for many different purposes. If you measure it in sufficient detail, you can construct a mathematical model of the structure that can be used in ...
2
Typically you are not doing modal analysis for its own sake. Typically you have a problem and you are using modal analysis as a tool to solve that problem. The goal of the modal analysis will depend on what exactly your problem is. Here are some examples
1) You have a part which is exposed to vibration, and is failing (cracking) due to high cycle fatigue. ...
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"Multiplicity" usually refers to several different modes with the same natural frequency. For example, ignoring any fault conditions and tolerances, if there are several identical planet gears, each one will have the same mode shapes and frequencies.
If you are measuring the vibration response, you may not be able to distinguish these modes - you will ...
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So, look at the time history that you added in edit 3. Notice how it is never ever ever below 1. That's very telling. Let us assume that you have a MEMS accelerometer is able to read down to 0 Hertz. So in the vertical direction, we expect to see 1g due to gravity PLUS the vibration. In other words, we expect to see $y(t) = 1 + A cos(\omega t)$, where A ...
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You can't make much use of just one FRF, especially if none of the peak responses line up between the model and the test.
Fix the accelerometer at the position of one of your model DOFs and then impact the model at all 12 DOF positions (in the correct directions, of course). Move the accelerometer to the other DOF positions in turn, and repeat.
You ...
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In the section "The Multiple DOF System" the Comsol document says
It has been assumed that the mass matrix normalization of the
eigenmodes is used and that the damping matrix
can be diagonalized by the eigenmodes.
Eigenvectors are assumed to be mass normalized in any mathematical derivation using them, unless somebody wants to be deliberately ...
1
Natural frequencies depend upon stiffness and mass. These two properties depend on the material characteristics and geometric configurations.
In general:
Increasing stiffness, the natural frequencies are increased
Increasing mass, the natural frequencies are reduced
With geometry things are a bit different. Because it will depend how you distribute mass ...
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We know for undamped harmonic vibration
$\omega= angular velocity \space, L=length of brick \ , m=mass \ , \\ I= \text{second moment of area of the brick section } \ , \ h= \text{the height of the brick section} $
$ \omega = \sqrt {k/m }$
And $ k=I/L $
$ I =bh^3/12$
So everything else being equal the thicker brick will have larger " I " hence larger ...
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Your "wall" would be referred as a "plate" in the technical literature. The eigenvalue problem for a plate is significantly more complicated than a beam, but similar ideas.
I would suggest that you look for a book on "vibrations of continuous systems", for example: http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471771716.html
There are also ...
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For a static problem, you can determine the displacements of a structure using only forces and stiffness of the structure using static equations for text books:
$F_{normal} = K \Delta \\ M_{bending} = EI \phi \\ T_{torsion} = GJ \psi $
Note that none of these equations depend on time. For the purpose of most problems in civil engineering, load application ...
1
I have only done this sort of optimization using non-commercial software.
Rather than trying to fit "the FRF" (i.e. the complete shape of all your measured response functions) I would start by estimating the frequencies and mode shapes from the measured FRFs and fitting just those parameters. In fact a good first attempt would be just to fit the frequencies,...
1
You are correct that the response for each mode must be calculated before applying some modal combination methodology.
Instead of thinking about the structure as being discretized into numerous elements, I like to think of it in terms of the lumped masses and how many DOFs there are. A more complex structure will simply have more DOFs (and therefore more ...
1
From my understanding, mass participation for each mode is determined
by the general mass and the eigenvector of the mode, and eigenvector
of the mode usually consists of contribution from all the $x$, $y$ and
$z$ direction, so how can one tell whether the mode is contributed
to lateral or vertical direction?
That is correct in general, but for a ...
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