Differential Equations By Np Bali Pdf

[Yi Pressimone]

Thecourse is to introduce the essential differential equations and their solution methods. The course is very much essential to all engineering students for its use in any kind of scientific or engineering work. The course offers them to good exposure of both ordinary and partial differential equations that arise in physical and engineering sciences.

Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.

The hyperbolic functions take a real argument called a hyperbolic angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector.

In complex analysis, the hyperbolic functions arise when applying the ordinary sine and cosine functions to an imaginary angle. The hyperbolic sine and the hyperbolic cosine are entire functions. As a result, the other hyperbolic functions are meromorphic in the whole complex plane.

Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert.[13] Riccati used Sc. and Cc. (sinus/cosinus circulare) to refer to circular functions and Sh. and Ch. (sinus/cosinus hyperbolico) to refer to hyperbolic functions. Lambert adopted the names, but altered the abbreviations to those used today.[14] The abbreviations sh, ch, th, cth are also currently used, depending on personal preference.

The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborn's rule[18] states that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) for θ \displaystyle \theta , 2 θ \displaystyle 2\theta , 3 θ \displaystyle 3\theta or θ \displaystyle \theta and φ \displaystyle \varphi into a hyperbolic identity, by expanding it completely in terms of integral powers of sines and cosines, changing sine to sinh and cosine to cosh, and switching the sign of every term containing a product of two sinhs.

Since the exponential function can be defined for any complex argument, we can also extend the definitions of the hyperbolic functions to complex arguments. The functions sinh z and cosh z are then holomorphic.

A fair warning: here be mathematics! While I try to avoid that at all cost, apparently this is a popular topic. I figured I will write on it, adding my anty-math twist to the mix! Take a look and let me know what do you think!

The above 2 equations are pretty easy to write in the matrix form. You just take the forces and put them in one vector, and you do the same with corresponding deformations. The rest is just matrix multiplication really:

We got the u2 deformation. But at this stage, we can also check if what we are getting makes any sense at all. Since u1 = 0, u2 is the deformation of node 2, but it will also be equal to total element elongation. If you remember the equation, you know that this is what we did above. But in any case, you can google for it, and check what the total elongation of our rod would be:

Imagine a steel column in a hall building with a crane. There is a roof supported by the column at the top (loading the column with 120kN of compression). Lower, the crane is operating and the load is 160kN at that level. The question is: what is the maximal load at the bottom of the column, and what is the vertical deformation on the top and crane level.

Finally, we can assemble the global stiffness matrix! Notice, that we have 4 DOF in our system so the matrix will be 44. Since we already know which elements contribute to which Degrees of Freedom (DOF) we can use all the above matrixes to create the global stiffness matrix:

Of course, we already know the coordinates. I will use the global ones here, assuming that x=0 is at the bottom of the column. This means that xi, being the beginning of element B will be at xi = 4m, the xj = 8m and the place we want (the middle of the element) x = 6m. With this in mind I can simply solve the above:

I have over 10 years of practical FEA experience (I'm running my own Engineering Consultancy), and I've been an academic teacher for a decade. Here, I gladly share my engineering knowledge through courses, and on the blog!

I always say that between "not knowing" and "knowing", the knowing is always better and more fun in engineering :)

It's just that there are more important things "to know" - although I do agree that understanding this all must be a great feeling for sure :)

Some of it comes from my head (like I've made sure that I understand the method, but I figured out the example myself - I just checked this in the proper FEA program to be sure I did it the right way!).

But I watched some lectures online on how the math is being done... and I think I just googled for "how to do X" when I wasn't sure. There is A LOT of material on this, and most of it is to my eye very similar - sadly I will be very hard-pressed to tell you which resources precisely I used, as I used them in a super minor way (like make sure if the equation is right, etc.)

But you can google FEA stiffness matrix, and you should find a lot of places offering help - this is a very popular topic it seems. However, all places I found focus on the "math" aspect, leaving the considerations in the "cruel twist" undiscussed... which is a pity I guess...

I really hate doing stuff like that and it actually hurts me to do so :P That being said, I know that folks may want to learn that (and I don't agree with the reasons behind this drive in most cases anyway)... but one day, I will sit on this too...

It's not that bad when you push through the initial "hurt"... but I won't lie - it is complicated. But this is a great FEA strength as well. Since it's a high barrier of entry things, people who master it are considered experts! There are benefits involved, so don't give up!

Does Young modulus equals 210e6=210x10^6 kPa?

The thing is I cannot get u2=0.317 mm when checking the result.

If E=210x10^6 kPa then I got u2=0.000317 mm

It seems there is something with units, but I cannot figure it out.

Please help.

Young Modulus is E = 210e9 Pa or 210e6 kPa... I left it as 210e6 kPa so I get the "kilo" in kiloNewtons. I could go with the 210e9 of course, but then the outcome would be in N/m not kN/m... I guess I've got a habit of calculating in kN :)

Of course, I've made the same mistake twice! Both when I calculated with FEA and by hand I've made the same mistake... after all both methods give the same outcome, that is 0.000317mm... yea that's math for me I guess :)

To be honest, I had to dig through several tutorials and even a book or two to do that... I don't feel like going through that once more, at least not right now :)

But it's obvious that you know what is going on - maybe you want to write something like that on the blog?

Thanks for your suggestion. In free time I will try to prepare something like that since I have some examples of simple plate FEM calculations from literature. It's not easy to find something like that in the books and even harder to find it in the internet articles so it would be nice to have it here. Such examples also show how structural analysis is based on partial differential equations because the stiffness matrix of 2D and 3D elements can't be derived from mechanical formulas like it's done for beams.

Yea, just be aware that I'm not super sure if we can just publish an example from a book. This is about copyright and all, so please make sure you are clear to prepare such an example... I don't want either of us to get in trouble because of this!

As for the books I used mostly the Nafems books (I'm a member so I have plenty of those) there is the thick hardcover one with basics (can't recall the title), it wasn't that much helpful (far too much theory with little to no examples) but it was a start : )

I don't think it will be useful with plates but maybe I'm just too stupid to see that, I don't know!

But it might be the case that I'm searching for stuff they were not meant to solve... I'm a very practical person not deeply interested in theory unless someone can show how it impacts the actual design...

The last paragraph and the last sentences were awesome , great

I remember , some month ago in 2018 , when I was 1st semester student of M.S Structural engineering , I had FEM course and the professor was so strict. it sends us problems , 10-times harder than the problem you solved in this post and I remember well the nights we started solving and when it finished , we understood the sun has rised

bad nights , bad memories

those days we never understood how FEM helps Civil engineers , Designers

those days we were full of bad feelings why we should learn something while we dont know its applications in Civil exactly , and how much should we swim in mathematical pools to get that ability to use FEM in our way

those days we were bored of reading Bhatti , Logan , Bathe and other books because of extreme pressure we were under it

I dont want those days and nights come back but I want to learn FEM once again , this time The Real & Useful FEM

Yea I know the pain. I admit on my Uni I had only to do simple cases by hand during Ph.D. studies (on a dynamics course to make it funnier). I'm not sure but if I remember correctly they were just demos, not more difficult than the example I did here...

I suppose the basics are kinda..needed? Especially if software informs you about "zeros on the stiffnes matrix diagonal", at least you know what this is ;). At our Uni, we have some "FEM in mathcad" and, while tedious, it helps in understanding what solver does.

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