Determine the centroidal moment of inertia, Mechanical Engineering

Determine the centroidal moment of inertia:

Determine the centroidal moment of inertia of a thin flat disc shown in Figure which is having a mass m kg. By using the parallel axis theorem, extend the result of cylinder of radius a and height h.

894_centroidal moment of inertia.jpg

Solution

We let the disc made up from rings of varying radius r and thickness dr as illustrated in the figure. The mass of each such ring per unit length shall be m¯ dr . The moment of inertia of this ring can be written as if dm = mass of one such ring then dm = 2πrdr m¯ .

=2 π m¯ a 4 /4

 = π a 2  m¯ .( a2/2)

Substituting m = π a 2 m¯ , total mass of the disc, we get

I zo= m .(a2/2)

and

= I xo = Iyo  = (1/ 4) ma2

and radius of gyration

kxo       = kyo = (½) a

The moment of inertia of the cylinder of Figure (b) about the axes x, y, z may be calculated by extending the above results. The cylinder may be imagined to consist of thin discs of thickness dz.

Suppose   γ = unit mass/volume

The total mass of the disc dm = π a 2  dz . γ . The moment of inertia of the disc about its own centroidal axes shall be

I zo = 1 dm . a 2  = 1 π γ a 4  dz

I xo       = I yo = (1 /4)π γ a 4  d z

Using parallel axes theorem, the moments of inertia of the disc around the axes x, y,

z are found as

I z = ∫ (½ ) π γ a 4  dz

I x = I y  = ∫(1/4) π γ a 4  dz + π γ a 2  z 2  dz

Integrating between the limits

z = - (½) h  to   z = + (½) h

gives,

 I z  = (γ π a 4  h )/2

I x = I y  = (¼) γ π a4   h + (1/12) γ π a2  h

The total mass of the cylinder m = γ π a 2  h , so the above results may be written as

I z  = ma2/2  , I x  = I y = (ma4/4) + ( 1/12)  mh 2

A sphere is having a radius R and mass m. calculate the moment of inertia about its diameter.

Posted Date: 1/29/2013 6:27:54 AM | Location : United States







Related Discussions:- Determine the centroidal moment of inertia, Assignment Help, Ask Question on Determine the centroidal moment of inertia, Get Answer, Expert's Help, Determine the centroidal moment of inertia Discussions

Write discussion on Determine the centroidal moment of inertia
Your posts are moderated
Related Questions
Define 3-D lathe Tool Dynamometer It is possible to measure three dimensional cutting tool component with the help of 3-D lathe tool dynamometer. In spite of that we are gone f

how to prove that work done is a path function

Orthographic Projections: As you learned previously about the theory of projection of points, planes and lines, the next significant is the projection of solids that have thre

Determine- cutting speed, Material removal rate and Machining time. A 150 mm long 12.7 mm diameter stainless steel rod is being turned to 12.19 mm diameter on a centre lathe.

when object is placed beyond the center of concave mirror


Two Possible Materials for Casing Two possible materials for the casing would be a. Polypropylene or b. An aluminium 1/3 per cent silicon-casting alloy (LM6). Conclu

The spring controlled governor of the Hartung type has two rotating masses each of 2.5kg and the limits of their radius of rotation are 100 mm and 125 mm. The each mass is directl

design a car to be build by balsa wood trusses, the car has to be with in a specific dimension, calculations of forces on members and nodes using method of joints need to be done a

Determine magnitude of the couple: A square block having each side 1.5 m is acted upon by system of forces along its sides as shown in the adjoining figure. If system reduces