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
An oil engine works on the Dual cycle, the heat liberated at consent pressure being twice that liberated at consent volume. The compression ratio of the engine is 8 and the expansi

Which of the following is an open pair?

do you help me with an assignment in machin dynamics which need a solid works program and CAD prototype

how can I determine the relative velocity in mechanism

Q. What is Expansion Joints? Expansion joints are used to accommodate thermal growth in piping and pressure vessel applications. The metal bellows, an integral part of an expan

The Dimension Style Command The Dimension Style command can be used to change appearance of dimensions. The best method  to create a new style before you can start creating dim

How to determine Forces and Torque of slider crank mechanism?

The following data were obtained in a boiler trail: Mass and temperature of feed water: 680 kg/hr and 20 0 C, Steam temperature and its pressure : 15bar and 300 0 C, Coal used

Find stress intensity factor - thick cylinder: A thick cylinder of r i = 10 mm, r 0 = 70 mm carried equal and symmetric radial cracks 6 mm long on opposite sides of inner su

Obtain the equations of the elastic curve and use them to determine the deflection and rotation at point B in terms of E, I, F and L.