24. Modal Analysis: Orthogonality, Mass Stiffness, Damping Matrix
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- Опубліковано 2 вер 2013
- MIT 2.003SC Engineering Dynamics, Fall 2011
View the complete course: ocw.mit.edu/2-003SCF11
Instructor: J. Kim Vandiver
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu
This is by far the most intuitive video about modal analysis I have found on the internet.
Thank you so much, Dr. Vandiver!
I feel like my tuition fees should really go to this course, since I've learned more here than I have in my actual lectures!
Share this video with anyone who is taking a course in mechanical vibrations or structural dynamics 👏 great explanations
As a civil engineering student, I really need this for my thesis. Thank you MIT Opencourseware.
DR. Vandiver, thank you for an excellent lecture on Modal analysis. This lecture is very practical and easy to follow.
Dr Vandiver is so thorough and delves way deeper than most dynamics courses..one other such course i know is by Calpoly prof on Intermediate Dynamics
The best modal lecture available ever, thank you so much.
Very good videos professor! I like this explanation
it is really changes my learning stratege in to beteer one thank you
Absolutely well done and definitely keep it up!!! 👍👍👍👍👍
Beautiful lecture professor.....seismic analysis
That was extremely useful. Thank you very much.
nice to go through it , thank you sir
Amazing lecture.
Great lecture!
It's not important for the point that he was making but at 12:45 the X in the first term of the equation of motion needs to be changed to X "double dot" (the second time derivative). The same applies to the expression at 13:45 where q needs to be changed to q "double dot".
omg , if we have such good professor, i will be already get my 2. doctor degree
Thank you so much!
Great lecture. thank you.
In the final 10-15 minutes we see that with proportional (Rayleigh) damping we obtain 2 parameters in order to approximate the modal damping ratios to the measured ones. How do we measure them in a 2dof system? And what is the point of finding damping ratios this way if we have them measured already?
Please help me.. how did he calculate omega 1 and omega 2 value as well as the mode shapes?
What is the difference between a natural mode and a mode shape?
thanks so much
Fantastic!
Good stuff
When does a structure have higher modes?
Please give an example of a structure that will have higher modes significant!
Awesome!!!
very explanatory
Gold
The mode shape matrix at 43:02 is not orthogonal. Am I missing something here?
They are orthogonal through the mass matrix or the stiffness matrix: U transposed x M x U = unit matrix
i also have same doubt
It is, the mode shape matrix houses the two eigenvectors that are in fact orthogonal to each other. You can't use the dot product to check this but instead the transpose of one of the eigenvectors* mass matrix* the second eigenvector. My answer on matlab was -5.1233*10^-4 which is close enough to 0
On 1:01:00 the professor says that "not alway equal to" but as far as I know for any orthogonal matrix u and any matrix c, that will always methamatically result into a diagonal matrix. Am I wrong?
The problem is that the vectors of the modal matrix U are not actually directly orthogonal despite what Prof Vandiver says in the earlier part of the lecture. The way to find the modal matrix is to solve for the eigenvectors of M⁻¹K. Note that if any mass terms are different from each other, M⁻¹K will not be symmetric. Although K is symmetric, M⁻¹ will be populated by the multiplicative inverses of each mass, and so different masses will scale each column of K differently, ruining symmetricity. Since M⁻¹K is not symmetric, then orthogonal eigenvectors are not guaranteed. A concrete example of a non orthogonal modal matrix is shown for an example at 42:58.
There is an alternate approach that explains why M and K are diagonalized by the modal matrix (and by extension, why C is not except for a special case). In this alternate approach, x is transformed twice - the first transform being x = M¹ᐟ²q. This transform decouples the modes so that you solve the eigenvectors for M⁻¹ᐟ²KM⁻¹ᐟ² instead (let's call it K̃), which is guaranteed to be symmetric (for any A and any symmetric B, AᵀBA is symmetric, and K must be symmetric, so M⁻¹ᐟ²KM⁻¹ᐟ² is symmetric), which in turn guarantees orthogonality of its eigenvectors. We then normalize those eigenvectors and assemble those eigenvectors into a matrix P. Normalizing the eigenvectors is critical so that P becomes orthogonal - a matrix composed of orthogonal columns is not orthogonal until they are normalized (so that P⁻¹P = I). Now, we can use a second transform (to diagonalize the left side matrices) q = Pr such that the left side of the equation becomes Pr̈ + K̃Pr, then left multiply the system by Pᵀ. Since Pᵀ = P⁻¹, and P is composed of the eigenvectors of K̃, P⁻¹K̃P = Λ by diagonalization, where Λ is a diagonal matrix (specifically, the spectral matrix, where the squared natural frequencies of each mode are the diagonal terms). The left side of the equation becomes Ir̈ + Λr.
Note that each term of UᵀMU is a scaled version of a term of I. This is not surprising, since I is the identity matrix. However, UᵀKU is a scaled version of Λ by the same terms. Consequently, any forcing is also scaled this way in his equations versus mine. Both of our methods would thus return the same result. Regardless, the entire reason why I went over this process was to show that the diagonalization process for M and K are unique to M and K. Including a damping matrix that does not commute with K will cause P to not be a basis of eigenvectors for C, which would cause it to not be diagonalized in the r-transform. Similarly, it will also not work using Prof Vandiver's method (UᵀCU).
how the orthoganality of a sine wave is zero?
first row first column entry is incorrect for u inverse correct value is .08978 at 1:17:10
at 45:14 how did you get the values of m1 and m2 and K1 and K2?
it was given in the problem. See at 34 min.
I was wondering why my stiffness matrix is not positive semidefinite for my beam, then I realized I need to fix some dofs, probably.
This is nice, interesting, and helpful, but the question that lingers for me is that, then, how do you calculate the natural frequencies and mode shapes of a damped MDOF system? Or the natural frequencies and mode shapes are for undamped systems only, which would mean that the natural frequencies and mode shapes of the damped system are those of its undamped version? Thank you.
I have the same doubt
Do u have an answer yet?
Hi. I know this is an old question but let me try to help others if not you.
A) If want to solve by hand and the system is lightly damped, just eliminate the damping from the EoM and you should be fine.
B) If you really want to solve the problem if the damping, keep in mind it would be very time-consuming. Basically, you would have a term [C]*dot{x} in your equation of motion, where the dot{x} is the x first time derivative. Due to that term, you'll get pairs of conjugated complex poles as roots of the Characteristic Equation for each mode shape. Your eigenvalues and eivenvectors will be complex.
B1) Although I believe there is more than one way to solve this kind of problem, the most used one is transforming the EoM to the State-Space form. Let's assume you're solving a 2 DoFs system. By doing the transform to state-space you're converting the set of two 2nd order differential equations to a set of four first order differential equations, by defining a state variable 'z'. Then you can write dot{z} = Az whereas A is a 4x4 matrix derived from the oritinal equations of motion.
B2) Then you can solve the Characteristic Equation: as for the undamped case, for each egenvalue you'll find the corresponding eigenvector.
B3) Formulate the general solution writing it as a combination of exponential functions based on the egenvalues and eigenvectors.
B4) Aplly the initial conditions to solve for the constants in the general form.
The state-space form is also very useful to solve damped problems in Matlab.
34.05 Mass m1=0.3193
Thank u sir
Step by step video solutions for civil engineering questions
which book is recommoned
Course materials including the suggested readings can be found in the full course site ocw.mit.edu/2-003SCF11. The readings are in the two following textbooks: Hibbeler: Engineering Mechanics: Dynamics www.amazon.com/exec/obidos/ASIN/0136077919/ref=nosim/mitopencourse-20 and Williams: Fundamentals of Applied Dynamics www.amazon.com/exec/obidos/ASIN/0471109371/ref=nosim/mitopencourse-20.
read L.Meirovitch or W.T. Thomson ( with or without p ) I can't say
For my previous education I used the book "Modal Testing: A Practitioner's Guide" from Peter Avitabile.
The book helped me with questions and practical tips for my custom vibration measurement services. Some of these are shown on my channel.
Is U always invertible?
The problem with a mathematical description of physics is that most people understand the physics before they understand the mathematics and when teaching it, you basically made it really difficult. You learn it in school and then you forget whereas a physical understanding is reinforced through physical observation of reality and most people don't forget just as we don't forget what it felt like to fall on our ass.
People fell on their arses for millennia before Newton and analysts following him, and had zero idea how any of this worked. They thought arrows were kept in the air by motions of the air. So, no.
I am having question regarding continuous system, plz help ,
when continuous system/MDOFs is vibrating,
how do we comes to know which modes are significant and till that mode sum to be taken?
+Chirag Palan Engineering intuition, thats what my professor told me. For simple problems you first just visualize the modes in your head and see if you can figure out breaking points, which can be done for simple shapes till like the fifth mode. Otherwise you need a analysis to figure out which modes are important.
in earthquake engineering the fundamental modes are those who are capable of 'activating' at least 80-90 of the total mass of the system when you add their particular contribution for each analysed direction . some approaches take into account those modes who move/activate at least 1%- 5%.
Very nice sir hi ji
DOUBT, (Need help...)
we know natural modes are linearly independent to each other (Orthogonal), then why the dot product of the natural modes is not zero in most of the videos (including this video).
Is it just a calculation mistake by the professor or my above mentioned concept is wrong?????
Any help will be appreciated ...
its funny when the professor realized his minor writing mistake.
Yeah, and he responded with the adequate attention
isn't this supposed to be "basic" ?
it is the basics of modal analysis, but i don't think we can call this basic ;)
Doff..equal modal ,§ j
19:00 Mic drop moment
no entiendo na
That inappropriate behavior of the students is the result of "democracy" in class, where you can come in what ever clothes you want, you can sit how ever you want, and at the end you can do what ever you want... Disrespect the knowledge and world will fall apart
it looks unfortunate that at the end of class, students do not care to let the teacher first come out of class before they leave the class room. This is slowly visible in other countries such as India also. The lecture is remarkable for its content and presentation.
a lot of math always a lot of math, no physical meaning discussion
igcr1234567890 ...Go to structural engineering! lol
Go cry dumbass
LOL if u really think that you haven't attended many engineering classes, this guy does a really good approach in both senses compared to many other proffessors that don't even understand physics behind this kind of problems
This is middle school level
My sentiments precisely! I had this 'experiment' with rope at grade seven. I've stopped watching when he missed double dots in the equation. Is it really MIT?
Anna B. I would say he used a 'simple' experiment to make the math sensible. lol If you try to work this stuff out, there's a lot of confusion of how terms turn out and it makes you feel very uneasy. If you demonstrate whatever problem you have with a simple experiment regardless of the level of education, it takes that uneasy feeling away slightly. Usually, I try to imagine how motion occurs all at once, but that's not always sufficient.
Not sure if joke..I didn't even see matrices in middle school lol
It is college level material. It is neither elementary nor complex, but a theory presented as it is.
Alex Poulin prolly just low quality bait. Surely if someone was taught this so early on and understood it they wouldn't look up introductory lectures on it right?