@tomleslie @JohnS 

Its been a fun ride, why cant "n" be anything else but an integer, its no fun.  alpha*L=n*Pi is a solution for the regid body beam theory, i was trying to achieve a solution for my specific scenario which is  valid for a G-G beam via K functions.  the missing link is the code and conditions i have to set to get the answer that was sought. higher end frequencies are very accurate with the method but for lower frequencies other methods should be considered. 

Anyway, if i wanted to consider a concentrated mass on the beam, how would i have to manipulate the PDE (govering equation) to consider that into account? Any jump start on that :D

Mech_Vib_Final.mw 

@tomleslie 

the final solution we got for W(x) was not in terms of sin, it was W(x) = D4*cos(n*Pi*x/L), and since D4 cannot be zero the cos term has to be zero but integers of n gives us the cos expression to be equal to 1 not zero :(

More over, what else is confusing me, why cant we plot a graph for it to find the roots(natural frequencies of the G-G beam)

the file i have attached has the solution implemented which i have solved via k-functions(characterictic eq was derived by hand in the notes i have attached). Another question, why is the solution different if solved by a different method and why does it plot the graph to give us the roots.

Im not saying the code is wrong or the answer achieved is wrong, since that is what youd get even if solved by hand, I cant seem to find a link between the two methods when they both solve for natural frequencies of the system.

K4square_equal_K2square1.mw

@acer 

if it were that easy, would you think id be crying here for a week? :D

@tomleslie 

so, times almost up, i have been searching everywhere to try to solve this, even talked to a couple of colleagues but they dont seem to want to help, even the professor has not replied to meet me since i messaged him last thurday. 

Anyway to cut the complications and imprecisions short, i need to plot for

(1/2(sinh(alpha*L)-sin(alpha*L))² = (1/2(sinh(alpha*L)-sin(alpha*L))²

I need to find the first 10 roots and/or solve for (alpha*L) which would give me the first 10 natural frequencies

I have attached the file, just need to plot for (alpha*L)

K4square_equal_K2square.mw

@tomleslie

Heres the code

Download Beam_vibration.mw 

@tomleslie 

I edited my reply above, i ran the exact code you uploaded, the reason i was getting the code because your simplified answer had the sinh and cosh terms combined and i was getting them seperately, i manipulated the subs(op) command and the error is gone, but i dont know how to replace (C1+C2)=D1 and (C1-C2)=D2 since all the subs(op) is doing is replace "C" constants with "D" constants

Sorry, my mistake, those were the asymptotic solutions for alpha*L

alpha*L= 0, 1*pi, 2*pi, 3*pi... (note: lets not take alpha*L=0 because that will give us a trivial solution which we dont want) therefore i suppose n is going to be 1,2,3... to find the roots for this specific solution?

@tomleslie 

Yeah you have solved for a non-trivial solution thats not what i was questioning, i was maybe thinking out loud on how to construct questions for the professor, alpha cannot be zero because that would make ω=0 which cannot be since we want to find ω for its first 10 natural frequencies. 

To solve for that we need to subsitute alpha*L = 4.73, 7.853, 10.996 to find the roots of the solution we got. I have been trying to solve it on Maply '15, but its throwing me errors since it said the code was constructed on a later version of maple. phi is also zero, we can remove it from the initial condition

Also, this is the error i get when i run it on my PC:

I removed phi from I.C and manipulated the sups(op) becuase my simpified answer was different than your, since i am using Maple '15 i think, anyway, how do i put (C1+C2)=D1 and (C1-C2)=D2

I will follow up with the professor about it, it seems good till the D2 and D4 which we have to solve simultaneously, i will refer to my notes and try to understand which conditions would be right for it to get a non-trivial solution. I also have to plot graph and see if the natural frequencies i get are coherent with the professors. Why did you take alpha*L=n*Pi, kinda confusing 

i implemented the BC, i let the solution stay as it is couldnt simplify it further but i am still confused what im missing, please tell me what im doing wrong

beam_vibration_displacement.mw