Torsional Vibration of a Rod

[sreid]

In Blevins "Formulas for Natural Frequency and Mode Shape" Table 8-19 gives Torsional Vibrations of Uniform Shafts.

C= Torsional Constant Table 8-18
Ip= Polar Area Moment of Inertia Table 5-1

In example 8.5.1 Tube Example, C is said to be equal to Ip but for a solid circular cross section Tables 8-18 and 5-1 seem to be in conflict.

The C formula is clear but what formula should I use for Ip (none of the formulas in 5-1 #20 match 8-18 #1).

 

[hacksaw]

Ip is the polar moment about the axis of the shaft whereas C is the moment about a principle axis...

 

[sreid]

So for a circular cross section the polar moment of inertia is R^2/2?

 

[desertfox]

Hi sreid

Moment of inertia for a rotating shaft according to my text book is:-

I= m*r^2/(2)

where m is the mass of the shaft and r is the radius.

regards desertfox

 

[sreid]

A new note-a dimensional analysis of the equation requires that Ip have the units of, say, inches^4 (not inches^2)sp R^2/2 is not correct.

 

[desertfox]

Hi sreid

Following your last post then you need the polar second moment of area

for a solid shaft j= 3.142*D^4/32

this j is normally used in formula for finding the shear stress in shafts subject to torsion ie:-


[τ]= T*r/(j)


where [τ]= shear stress

T= torque on shaft

r= shaft radius

j= polar second moment of area

hope this helps

regards desertfox

 

[sreid]

Desertfox,

Thanks. I carefully re-read the introduction and the "C" and "Ip" are identical for circular sections and cancel out since they are in the equation in the form "C/Ip" (as incredible as that seems).

 

[GregLocock]

sreid, look at table 8-18.






Cheers

Greg Locock

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