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Okay, so good morning everyone and I guess

this is the last topic of the course

and I'm not sure if this is the last class.

Well, obviously yes,

but it's possible that I may not be able to cover

some of it in the finalцен,

but the important thing to note that

it's required by the new

Copyra

to cover some of it. So, I might take one extra class if you want to come you are welcome,

but whatever I take in the extra class that will not be covered in the exam for sure.

Alright, so as I said to you in the last class, we want to discuss finite element discretization

of equations of special cosrots rods. So, finite element discretization of equations

of special cosrots rods. So, remember you we already learned in the class how to solve

the equations of special cosrots rods where we wrote the equations as a system of 13 first

order ODE's right and we posed it as an boundary value problem which could be solved using

commercial packets for example, MATLAB right. Now, the today's technique is a different

technique we would solve it using finite element methods ok.

So, as you all know what is the first step when you want to solve anything using FEM

finite element methods. So, we write down the strong form of the equations and then

we derive the weak form of it. So, that is the first step ok. So, the strong form of

the equations are n prime plus n hat equal to rho naught a little r double dot ok. That

is the linear momentum balance and the angular momentum balance is m prime plus little r

prime cross n plus m hat that is equal to rho naught times the moment of area tensor

i into angular velocity omega and the whole thing and then dotted time derivative of that

ok. So, that is the angular momentum of the cross section and then we take it is time

derivative that gives you the rate of change of angular momentum of the cross section ok.

So, this is the strong form of the equation and when you want to get the weak form we

multiply the strong form with test functions right or the variations. So, to the linear

momentum balance we are going to dot this equation with del r where del r is nothing

but the variation in r u little r and similarly to the angular momentum balance we are going

to dot it with del theta and again this is the test function or variation corresponding

to change in the variable theta ok. So, that is the test function slash variation corresponding

to little r and this is the same thing but corresponding

to theta ok and the variable theta gives you the rotation matrix how that give you? You

simply say the rotation matrix big R is equal to exponential of the skew symmetric tensor

theta ok. So, essentially this rotation although it is a 3 by 3 matrix it has only 3 unknowns

in this theta vector ok. So, that is the axis angle representation find the skew symmetric

matrix of theta exponentiate it then you get your rotation tensor alright. So, little r

and theta are our kinematic unknown variables for special cos rad rods and del r and del

theta are the corresponding variations in those variables alright.

So, once we dot it with the test function and then we also do the integration over the

length of the rod. So, it goes from 0 to l and this also goes from 0 to l ok and then

we write it write them together. So, this becomes rho naught a little r double dot dot

del r plus rho naught i omega dot dot del theta ok. So, I write this separately this

is the dynamic term. So, I write the dynamic term separately that goes from 0 to l d s

plus the remaining term now bring them on the other side these things. So, it comes

with a minus sign then minus 0 to l n prime plus n hat dot with del r and one more minus