Why is this rigidity constant decomposed using a squared cosine?

[Guidestone]

Good night guys. I've been studying for a test on mechanical vibrations and I came across this problem of a telescopic crane (or whatever its name is). It's necessary to obtain a equivalent constant kp and then it has to be decomposed into its vertical component and the author uses a squared cosine of 45 degrees to do so. I would have used a sine of 45 degrees instead without squaring it. I know the sine and the cosine have both the same result, I'm just using a sine because the vertical component is needed.
Could you please tell me why the author is using a squared cosine? Also, why is the force pulling and not pushing when the weight of the guy is obviously going downwards?
Thank you

 

[Dr.D]

Off the top of my head, I don't have a good answer to your question. That said, let me note that the axial vibration is rarely ever the major concern for a system such as this. In reality, it needs to be treated as a cantilever in bending, with non-uniform bending stiffness. Bending is almost always the dominant deformation mode when both axial and bending deformations are present. I'm going to have to think a while longer about your original question.

 

[Guidestone]

Yes please Dr.D. thanks

 

[Guidestone]

By the way. I'm sorry if this was posted in the wrong forum. I'll be posting this kind of threads in the homework section in the future. Thanks for your understanding.

 

[Dr.D]

The problem statement appears to be in Spanish (or is it Portuguese?), which just about wipes me out. I can guess at what the problem is asking, but based on that, the answer given makes no sense at all. To do much better, I would suggest that you translate the problem statement into English. After that, I and probably others as well, will be able to do more to help you.

 

[Guidestone]

Yes, I will translate it. Thank You. :)

 

[Dr.D]

The attached PDF gives an explanation of this problem that I hope will make clear the very limited role that the boom angle plays.

 

[JBA]

As I translate it, the first section of the problem is simply a calculation of the composite kb (stiffness) for the combined three boom segments' k1, k2, k3 along the axis of the boom, ignoring the bending of the beam; with a following conversion of that linear beam kb to vertical component k. As a result, I cannot see any reason for the use of cos^2 45° rather than simply sin 45°.

 

[Dr.D]

Request to Guidestone:
May we have a citation for the book where this problem is found (author, title, publisher, date, page number), please? Also, is the language Spanish?

 

[Guidestone]

Hi, thank you for the answers and the material you have posted. I've been really busy with exams so I don't have the chance to be around. The book's name is mechanical vibrations and it was written by Singiresu S.Rao. I will make my best to get the English version of it. I don't remember the number of the page but I'll make it available as soon as I finish with school stuff. The language is Spanish indeed. :)
Thanks again

 

[Dr.D]

Guidestone, I look forward to your translation and the page number (don't forget the publisher, date, and edition also). Thanks,