Moment of Inertia and Centroid

Transcription

1 Chapter- Moment of nertia and Centroid Page- 1. Moment of nertia and Centroid Theory at a Glance (for ES, GATE, PSU).1 Centre of gravity: The centre of gravity of a body defined as the point through which the whole weight of a body may be assumed to act.. Centroid or Centre of area: The centroid or centre of area is defined as the point where the whole area of the figure is assumed to be concentrated.. Moment of nertia (MO) About any point the product of the force and the perpendicular distance between them is known as moment of a force or first moment of force. This first moment is again multiplied by the perpendicular distance between them to obtain second moment of force. n the same way if we consider the area of the figure it is called second moment of area or area moment of inertia and if we consider the mass of a body it is called second moment of mass or mass moment of nertia. Mass moment of inertia is the measure of resistance of the body to rotation and forms the basis of dynamics of rigid bodies. Area moment of nertia is the measure of resistance to bending and forms the basis of strength of materials.. Mass moment of nertia (MO) mr i i i Notice that the moment of inertia depends on the distribution of mass in the system. The further the mass is from the rotation axis, the bigger the moment of inertia. For a given object, the moment of inertia depends on where we choose the rotation axis.

2 Chapter- Moment of nertia and Centroid Page- n rotational dynamics, the moment of inertia appears in the same way that mass m does in linear dynamics. Solid disk or cylinder of mass M and radius R, about perpendicular axis through its centre, 1 MR Solid sphere of mass M and radius R, about an axis through its centre, /5 M R Thin rod of mass M and length L, about a perpendicular axis through its centre. 1 1 ML Thin rod of mass M and length L, about a perpendicular axis through its end. 1 ML.5 Area Moment of nertia (MO) or Second moment of area To find the centroid of an area by the first moment of the area about an axis was determined ( x da ) ntegral of the second moment of area is called moment of inertia ( x da) Consider the area ( A ) By definition, the moment of inertia of the differential area about the x and y axes are d and d d y da y da d x da x da

3 Chapter- Moment of nertia and Centroid Page-.6 Parallel axis theorem for an area: The rotational inertia about any axis is the sum of second moment of inertia about a parallel axis through the C.G and total area of the body times square of the distance between the axes. NN CG + Ah.7 Perpendicular axis theorem for an area: f x, y & z are mutually perpendicular axes as shown, then zz ( J) + Z-axis is perpendicular to the plane of x-y and vertical to this page as shown in figure. To find the moment of inertia of the differential area about the pole (point of origin) or z- axis, (r) is used. (r) is the perpendicular distance from the pole to da for the entire area J r da (x + y )da + (since r x + y ) Where, J polar moment of inertia.8 Moments of nertia (area) of some common area: (i) MO of Rectangular area Moment of inertia about axis XX which passes through centroid. Take an element of width dy at a distance y from XX axis. Area of the element (da) b dy. and Moment of nertia of the element about XX axis da y b.y.dy Total MO about XX axis (Note it is area moment of nertia) + h h bh by dy by dy 1 h 0

4 Chapter- Moment of nertia and Centroid Page- Similarly, we may find, hb 1 Polar moment of inertia (J) + bh hb bh 1 f we want to know the MO about an axis NN passing through the bottom edge or top edge. Axis XX and NN are parallel and at a distance h/. Therefore NN + Area (distance) bh h bh + b h 1 Case-: Square area a 1 Case-: Square area with diagonal as axis a 1

5 Chapter- Moment of nertia and Centroid Page- 5 Case-: Rectangular area with a centrally rectangular hole Moment of inertia of the area moment of inertia of BG rectangle moment of inertia of SMALL rectangle BH bh 1 1.(ii) MO of a Circular area The moment of inertia about axis XX this passes through the centroid. t is very easy to find polar moment of inertia about point O. Take an element of width dr at a distance r from centre. Therefore, the moment of inertia of this element about polar axis d(j) d( + ) area of ring or d(j) π rdr r (radius) ntegrating both side we get R πr πd J π r dr 0 Due to summetry Therefore, 6 J πd πd πd and J 6

6 Chapter- Moment of nertia and Centroid Page- 6 Case : Moment of inertia of a circular area with a concentric hole. Moment of inertia of the area moment of inertia of BG circle moment of inertia of SMALL circle. πd - π d 6 6 π 6 π (D d ) and J (D d ) Case : Moment of inertia of a semi-circular area. 1 NN of the momemt of total circular lamina 1 πd πd 6 18 We know that distance of CG from base is r D hsay ( ) π π i.e. distance of parallel axis XX and NN is (h) According to parallel axis theory NN ( ) + Area distance G πd 1 πd or + ( h) 18 πd 1 πd D or + 18 π or 0.11R

7 Chapter- Moment of nertia and Centroid Page- 7 Case : Quarter circle area XX one half of the moment of nertia of the Semicircular area about XX. 1 XX ( 0.11R ) R XX 0.055R NN one half of the moment of nertia of the Semicircular area about NN. 1 πd πd NN 6 18 (iii) Moment of nertia of a Triangular area (a) Moment of nertia of a Triangular area of a axis XX parallel to base and passes through C.G. XX bh 6 (b) Moment of inertia of a triangle about an axis passes through base NN bh 1

8 Chapter- Moment of nertia and Centroid Page- 8 (iv) Moment of inertia of a thin circular ring Polar moment of nertia ( ) J R area of whole ring R πrt πr t J XX YY πr t (v) Moment of inertia of a elliptical area XX πab Let us take an example: An -section beam of 100 mm wide, 150 mm depth flange and web of thickness 0 mm is used in a structure of length 5 m. Determine the Moment of nertia (of area) of cross-section of the beam. Answer: Carefully observe the figure below. t has sections with symmetry about the neutral axis.

9 Chapter- Moment of nertia and Centroid Page- 9 We may use standard value for a rectangle about an axis passes through centroid. i.e. bh. The section can 1 thus be divided into convenient rectangles for each of which the neutral axis passes the centroid. - Beam Rectangle Shaded area ( ) m m.9 Radius of gyration: Consider area A with moment of inertia. magine that the area is concentrated in a thin strip parallel to the x axis with equivalent. k A or k A k radius of gyration with respect to the x axis. Similarly k A or k A J ko A or ko k k + k o J A Let us take an example: Find radius of gyration for a circular area of diameter d about central axis. Answer:

10 Chapter- Moment of nertia and Centroid Page- 10 We know that, K A or K π d 6 d A πd XX XX

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