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MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY
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MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Dec 30, 2015

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Job Singleton
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Page 1: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

MOMENT OF INERTIA

BY GP CAPT NC CHATTOPADHYAY

Page 2: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

WHAT IS MOMENT OF INERTIA?

IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION

IT IS RESISTANCE OF BENDING MOMENT OF A BEAM

IT IS THE SECOND MOMENT OF MASS (mr2) OR SECOND MOMENT OF AREA (Ar2)

IT’S UNIT IS m4 OR kgm2

Page 3: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

PERPENDICULAR AXIS THEOREM

    The moment of inertia of a plane area about an axis normal to the plane is equal to the sum of the moments of inertia about any two mutually perpendicular axes lying in the plane and passing through the given axis.  Moment of Inertia: Iz = Ix+Iy

Page 4: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

PARALLEL (TRANSFER)AXIS THEOREM

THE MOMENT OF AREA OF AN OBJECT ABOUT ANY AXIS PARALLEL TO THE CENTROIDAL AXIS IS THE SUM OF MI ABOUT IT’S CENTRODAL AXIS AND THE PRODUCT OF AREA WITH THE SQUARE OF DISTANCE OF CG FROM THE REF AXIS

IXX= IG+Ad2

A is the cross-sectional area. : is the perpendicuar distance between the centroidal axis and the parallel axis.

Page 5: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Moment of Inertia - Parallel Axis Theorem

Parallel axis theorem:

Consider the moment of inertia Ix of an area A with respect to an axis AA’. Denote by y the distance from an element of area dA to AA’.

2xI y dA

Page 6: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Moment of Inertia - Parallel Axis Theorem

Consider an axis BB’ parallel to AA’ through the centroid C of the area, known as the centroidal axis. The equation of the moment inertia becomes

22x

2 22

I y dA y d dA

y dA y dA d dA

Page 7: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Moment of Inertia - Parallel Axis Theorem

The first integral is the moment of inertia about the centroid.

0

0

y A y dA y

y dA

2xI y dA

The second component is the first moment area about the centroid

Page 8: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Moment of Inertia - Parallel Axis Theorem

Modify the equation obtained with the parallel axis theorem.

2x 2I y dA y dA 2

2x

d dA

I d A

Page 9: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Example – Moment of Inertia

Compute the moment of inertia in the x about the AA` plane.

AA`

Page 10: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Example – Moment of Inertia

Compute the moment of inertia in the x about the AA` plane.

AA`

h b2 2

x

Area 0 0

h3 3

03 3

I y dA y dxdy

y bhb

Page 11: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Example – Moment of Inertia

From earlier lecture, the moment of inertia about the centroid

h/22 2

x

Area h/2

h/23 3

h/23 12

I y dA y bdy

y bhb

Page 12: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Example – Moment of Inertia

Using the parallel axis theorem

2x x

233

3

4

12 2 12

3

I I d A

bh hbh bh

bh

AA`

Page 13: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Parallel Axis - Why?

Recall that the method of finding centroids of composite bodies? -

Follow a Table technique

How would you be able to find the moment of inertia of the body. Use a similar technique, table method, to find the moment of inertia of the body.

Page 14: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Parallel Axis - Why?

Use a similar technique, table method, to find the moment of inertia of the body.

Bodies Ai yi yi*Ai Ii di=yi-ybar di2Ai

i i

i

y Ay

A

2x x

i

2

xi i i

I I d A

I y y A

Page 15: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Moment of Inertia

Use a set of standard tables:

Page 16: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Example - Moment of Inertia

Find the moment of inertia of the body, Ix and the radius of gyration, kx (rx)

Page 17: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

I THINK…. I CAN THINK….

Page 18: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Example - Moment of Inertia

Set up the reference axis at AB and find the centroidBodies Ai yi yi*Ai Ii di=yi-ybar di

2Ai

1 18 1 182 18 5 90

36 108

3i i

2i

108 in3.0 in.

36 in

y Ay

A

Page 19: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Example - Moment of Inertia

From the table to find the moment of inertia

2

x xi i i

4 4 460 in 144 in 204 in

I I y y A

Bodies A i y i y i*A i Ii di=y i-ybar di2A i

1 18 1 18 6 -2 722 18 5 90 54 2 72

36 108 60 144

ybar 3 in.

I 204 in4

Page 20: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Example - Moment of Inertia

Compute the radius of gyration, rx.

4x

x 2

204 in

36 in2.38 in.

Ir

A

Page 21: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Example - Moment of Inertia

Find the moment of inertia of the body, Ix and the radius of gyration, kx (rx)

Page 22: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Example - Moment of Inertia

The components of the two bodies and subtract the center area from the total area.

Bodies Ai yi yi*Ai Ii di=yi-ybar di2Ai

1 21600 90 1944000 58320000 0 02 -9600 90 -864000 -11520000 0 0

12000 1080000 46800000 0

ybar 90 mm

I 46800000 mm4

Page 23: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

Example - Moment of Inertia

Compute the radius of gyration, rx.

4x

x 2

46800000 mm

12000 mm62.45 mm

Ir

A

Page 24: MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY. WHAT IS MOMENT OF INERTIA? IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION.

PROBLEM….

FIND THE MI OF A CIRCULAR LAMINA OF RADIUS “r“?