Why is this rigidity constant decomposed using a squared cosine?

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Discussion Overview

The discussion revolves around a problem related to mechanical vibrations, specifically concerning the decomposition of a rigidity constant in a telescopic crane scenario. Participants explore the use of a squared cosine function in this context and the implications of force direction in relation to weight.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant questions the author's use of a squared cosine of 45 degrees for decomposing the rigidity constant, suggesting that a sine of 45 degrees would suffice for obtaining the vertical component.
  • Another participant notes that axial vibration is not typically the primary concern for such systems, emphasizing the need to consider bending as the dominant deformation mode.
  • A participant expresses confusion about the problem statement being in Spanish or Portuguese, indicating that this hampers their ability to assist effectively.
  • One participant mentions that the problem involves calculating the composite stiffness of three boom segments and questions the necessity of using cos^2 45° instead of sin 45°.
  • There is a request for citation details regarding the source of the problem to facilitate further discussion.

Areas of Agreement / Disagreement

Participants express differing views on the appropriate mathematical approach for decomposing the rigidity constant, with no consensus reached on the necessity of using a squared cosine function. Additionally, there is uncertainty regarding the language of the problem statement, which affects the clarity of the discussion.

Contextual Notes

The discussion highlights limitations related to language barriers and the need for translation to fully engage with the problem. There are also unresolved assumptions about the role of bending versus axial deformation in the context of the crane's mechanics.

Guidestone
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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
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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.
 
Yes please Dr.D. thanks
 
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.
 
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.
 
Yes, I will translate it. Thank You. :)
 
The attached PDF gives an explanation of this problem that I hope will make clear the very limited role that the boom angle plays.
 

Attachments

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°.
 
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?
 
  • #10
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
 
  • #11
Guidestone, I look forward to your translation and the page number (don't forget the publisher, date, and edition also). Thanks,
 

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