Deriving Equations of Motion for Two-Step Cone Pulley Vibration

[toolpusher123]

Hi, I'm currently studying Mechanical Eng & have a class next semester 'Eng Analysis', which involves a large section on vibration. I've been trying to work my way through S.S.Rao, Mech Vibrations, 2nd edition. There's one specific problem; Project 5.51/Chapter 5/page 292.
The problem concerns; two step cone pulleys connected by a belt. All relevant variables are given i.e. input/output shaft rpm, driven puley rpms, distance between centres,polar moments of inertia etc.
I understand that this is a 2 Degrees of Freedom problem, & I should draw a free body diagram ( 2 pulleys connected by 2 springs). It's theinitial 'Equations of Motion' that I can't derive, & without these I can't really start the problem. I've attached a screen shot containing a clear example of the problem.
Any help that's offered would be greatfully received.

 

[berkeman]

Hi, I'm currently studying Mechanical Eng & have a class next semester 'Eng Analysis', which involves a large section on vibration. I've been trying to work my way through S.S.Rao, Mech Vibrations, 2nd edition. There's one specific problem; Project 5.51/Chapter 5/page 292.
The problem concerns; two step cone pulleys connected by a belt. All relevant variables are given i.e. input/output shaft rpm, driven puley rpms, distance between centres,polar moments of inertia etc.
I understand that this is a 2 Degrees of Freedom problem, & I should draw a free body diagram ( 2 pulleys connected by 2 springs). It's theinitial 'Equations of Motion' that I can't derive, & without these I can't really start the problem. I've attached a screen shot containing a clear example of the problem.
Any help that's offered would be greatfully received.

What does the textbook say about this kind of problem? What equations does the text use when describing this kind of problem?

 

[AlephZero]

From the question:

Input speed 350 RPM, output speed 150 RPM
Input diameter 250mm, output diameter 1000mm

? Either I don't understand the question, or there is a typo somewhere. Those numbers don't seem to be consistent with each other.

 

[toolpusher123]

Same problem, updated rpm's & diameters...

Hi, sorry here's the question with updated variables i.e. input shaft speed, output pulley speeds.
Young's modulus for the belt: j1= 1x10^10 N/m^2, Drive step cone mass moment of inertia: j2=0.35 kg.m^2, Driven step cone mass moment of inertia: 0.6 kg.m^2, Distance between centres: c=2.8m, belt width: 19mm
Please see attached for details.

1st step: Free Body Diagram. this is essentially the 2 masses connected by 2 springs.
2nd step: Using Newton's 2nd law (∑F=ma), write Eq's of motion.
3rd: Assume simple harmonics. x=X sin (ωt+ø), x'=ωX cos (ωt+ø)
4th: determine natural frequencies...
I think it's 'Rayleigh's energy method', I should use to find the KE of the step cones & PE of the springs.
It's the initial Eq's of motion that I can't derive.

 

[toolpusher123]

N=300rpm
No1=650rpm
No2=450rpm
No3=250rpm
No4=100rpm
d1=400mm
d2=300mm
d3=200mm
d4=100mm

 

[toolpusher123]

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