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γ is a coefficient that quantifies the extent of damping relative to the internal elastic forces. This is a quantity that is typically determined in the lab and is a function of the material of the beam, its geometry and the boundary conditions and the type of excitation. Typical values range from about 0.01 to 0.1. The specific reference I used for the is "Dynamics of Structures" by Hurty & Rubinstein. The book is long since out of print, but you can find a copy at archive.org. Not sure how helpful you'll find it though. For a reference on variational principles, my go-to reference is "Structural Dynamics: A Variational Approach" by Dym & Shames which is an excellent book! You'll likely also find an archived copy somewhere online.

@@Freeball99 thank you very much sir! i'll keep watching

This is the fundamental lemma of the calculus of variations and it is essential that you are clear on this! The idea is that if I have Int f(x)·g(x) dx = 0 over some domain and we know that g(x) is arbitrary (it could be any value at any point within the domain), then f(x) must be zero AT EVERY POINT in the domain. For example, if g(x) were zero at every point in the domain except at one point and at that point f(x) is also non-zero, then it would be impossible to make this integral equal zero. The only solution then is that f(x) is zero at every point.

I'm not sure I understand the question. Where did this expression above come from? Do you have a time code for me to reference? Are you asking about what the different component of the stiffness matrix represent?

@@Freeball99 I'm asking about the middle node. Force for the middle node as given by k*u is the expression I gave. But I'm not sure what kind of force to represent with that expression. Is it nodal force? Internal force? Or what?

@nedisawegoyogya {F} = [K]{q} where the [K] has been derived for you in the video and q are the nodal displacements. This gives the forces at each node, so F2 - ie the middle row from the vector equation above, will give the force at node 2. Note U is a function of x and t while q is a function of t only, so q can be taken out of the integral.

@@Freeball99 can you please give the integral form for F2, and explain why the integral represent the force with fundamental law? Because K itself doesn't explain a lot in terms of representability, because it came from integral

@@Freeball99 of course by integral form I mean no vector only variables for the scalar F2

Don't think I will cover Hamiltonian mechanics in this series because it is outside the scope of this material. Hamiltonian mechanics are really not used much in structural dynamics (as it is not well-suited to structural dynamics type problem - especially those in which energy is dissipated), but rather tends to be used in quantum mechanics and control theory.

The app is "Paper" by WeTransfer. Running on an iPad Pro 13 inch and using an Apple Pencil.

Not really related. However, both allow one to extract the fundamental frequencies from the system, so they have that in common. In this particular video, we are dealing with ODEs and not a PDEs. We are using the normal mode method to decouple the equations of motions which allows us to easily solve each equation separately. We do this in the time domain. Fourier transform is used to transform a system from the time domain to the frequency domain.

"Solid Mechanics: Variational Approach" by Dym & Shames tends to be my go-to book for this sort of material. www.google.com/books/edition/Solid_Mechanics/rTw_AAAAQBAJ?hl=en&gbpv=1&printsec=frontcover

At this time, I am examining the particular solution... Eqn 1 contains x and x_ddot. Eqns 5 & 7 give me expressions for x and x_ddot (I don't need eqn 6) I substitute these two expressions into eqn 1, expand it out, then take out cos ωt and sin ωt as common factors.

Very kind of you. Didn't expect you to still answer questions years after your good work. I am a medical doctor self learning this information to try to understand how the human body works. And I believe vibration represents the common, most fundamental manifestation of all life forms. All measurable parameters of the human body such as PH, blood flow, oxygenation etc all finally distillate to the ability of the cells to vibrate. I believe the human body runs along the principle of least action, and eigen format. The systems inside us do not procrastinate like we do! They are very measurable and predictable. Engineering knowledge has really opened my eye understanding how medicine works. Thank you for your kind attention. Wish I had an academic buddy like you for brainstorming.

Food for thought...Every piece of matter in the universe that is above zero Kelvin has thermal energy. This causes its molecules to vibrate. So vibration certainly is the "most fundamental manifestation of all life forms" BUT ALSO of all non-life forms. Literally everything physical. A definition of 0 Kelvin is that it is the temperature at which all molecular motions ceases.

@@Freeball99 You are quite right sir. A dead cell or dead body still vibrates according to thermodynamic principles. But I believe the innate natural frequency will be different compared to when it is alive. All the metabolic, physiologic and chemical activities in living things determine the natural frequency when the creature is alive. My intention is to study these frequencies with engineering method such as solving for eigen vector or applying a high frequency signal and try to detect from the output any frequency spikes. While metabolic resonance may run the risk of stimulating cancer growth, I hope by altering the physical parameters of an organ such as compressing it or stretching it, such as in the case of a kidney, so that different " natural frequencies" can be created to the same organ, by subjecting the organs to resonante at different frequencies, we can augment the functioning on the one hand, and destroy any unwanted entities such as worms, virus or even cancer cells within the organ. Thank you for your interest and appreciate your professional wisdoms. Please keep in touch.

I discuss linearity around the 10:45 mark.

Delta's occurs in many places in math and engineering. In this case, we are referring to the variational operator. It bears no relation other uses like the Dirac delta function or the Kronecker delta.

In this problem, I have not assumed any gravity is present. I have assumed that the external load is some general function so if you wanted to include the weight of the beam, you could include it in the external load.

Yes, the sign of the last 2 terms of eqn 25 should both be negative (careless error), however, since we're setting each of these terms to zero, it makes no difference in the final analysis and yes these terms should be part of the integrand of the time integral (the dt should be on the line below). Thanks for catching that.

There you go: ruclips.net/video/mecThHHV4eg/видео.html

Yes, it is exactly the same. When deriving things in this video, I glossed over some of the formality in deriving it because I wanted to keep it simple, but then in the Timoshenko video, I wanted to lay it all out since I expect viewers of that video to be a little more familiar with the material. The only difference between the 2 theories is in the strain energies, since the strain energy for the Timoshenko beam includes shear and the EB beam does not. This is consistent with the EB assumption that cross-sections that are normal to the elastic axis before deformation remain normal to the elastic axis after deformation.

@@Freeball99 thank you for your reply. I found your video that actually did it (deriving Bernoulli equation of motion from Hamilton), so actually it already answered my original question. Thanks again anyway for explaining it again. Your videos are really awesome.

I skipped a step here. Since L = T - V, if I substitute this into Lagrange's Equation, then it reduces to the form I have in the video. It reduces to this because in this problem, the kinetic energy, T, is a function of q_dot only and does not explicitly depend on q.

1. It's not the derivative ∂y‘, but rather the variation 𝛿y' that we are integrating. The formula for integration by parts is: int u dv = uv - int v du. So, in our problem, dv = 𝛿y' therefore integrating gives us v = 𝛿y. Then we plug this into the formula. 2. This is how we integrate by parts. Just plugging into the formula above. Perhaps try to review this technique. 3. The reason we say that the entire expression must be equal to zero is as follows... - If the value of the integral, I, is to be a maximum or minimum, then the integral must not change its sign for all possible variations of 𝛿y. - However 𝛿y is arbitrary (ie can be positive or negative), therefore the part that multiplies 𝛿y equal to 0. - We conclude from this that 𝛿I = 0 is the necessary condition to find an extremal. Hope this makes sense.

Short Answer: you can now ask Chat GPT for the best substitution to use. Longer Answer: the form of the denominator sqrt(c - y) gives us the clue that a trig substitution is the way to go. From there, you can find a table of useful trig substitutions and look for one with the same form as the integral that we have - ie. sqrt(x) / sqrt(a - x). Old School Alternative: Use a table of integrals to integrate this.

Not sure what I was thinking when I wrote this. As I look at equation 20 now, it seems correct to me. Will edit my comment. Thanks for the feedback.