Principal Moments of Inertia

One of the major interest in the moment of inertia of area A is determining the orientation of the orthogonal

axes passing a pole on the area with maximum or minimum moment of inertia about the axes.

Product of Inertia

Similar to the moment of inertia, a product of inertia can also be obtained from an integral over an area by

multiplying the product of the coordinates x and y about the reference coordinate axes by the elemental area.

Imply:

When considering the second moment of an area as the effect of the first moment acting on the same reference

axis, the product moment of an area can be considered as the cross effect of the first moment acting on the

orthogonal axis through a origin O at the specified orientation with respect to the area A.

Unlike the moment of inertia, although the elemental area is positive, the product of inertia can be positive,

negative, or zero because the value of the coordinates x and y can be positive, negative, or zero. Similar to the

first moment of an area about the the axis of symmetry, when one or both of the coordinate axes, x and y are

the axis of symmetry of the area A, the integral, the product of inertia Ixy about the coordinate axes is zero.

For example, a symmetrical area:

Although the area A is not symmetrical about axis y, however since the area is symmetrical about axis x, for

any elemental area at a distance y above the axis x, there is always an elemental area below the axis x at the

same mirror location of distance -y below the axis x. Therefore the product of inertia of a paired elemental

area will cancel out each other and becomes zero, and the integral will reduces to zero also. Imply:

Since the product of inertia of a symmetrical area about one or two axes of symmetry must be zero, the product

of inertia of an area with respect to axes can be used to test the dissymmetry or imbalance of the area about x

and y axes because when the product of inertia about x and y axes is not equal to zero, the area is not

symmetrical about both x and y axes. But when the product of inertia about x and y axes is equal to zero, the

area may be not symmetrical about x and y axes.

Parallel-Axis Theorem of Product of Inertia

Assume axes x and y are the interested rectangular coordinate axes and axes x' and y' are the two rectangular

centroidal axes of the area parallel to the coordinate axes respectively. Using axes x and y as the system

coordinate, the coordinate of the centroid C can be denoted by x and y accordingly and the coordinate of the

elemental area can be denoted by x and y respectively. Therefore the location (x,y) of the elemental area can

also be expressed in terms of the coordinate (x,y) of the centroid C as derived in the parallel-axes theorem for

the second moment of an area. Imply:

The first integral is the product of inertia Ix'y' of the area A about the centroidal axes, x' and y' . The second

an third integrals are the first moment of the area A about the centroidal axes, x' and y' and the value of the

first moment of the area about the centroidal axis is equal to zero. The fourth integral is equal to the total area

only. Imply:

Similarly, both the coordinate (x,y) of centroid C of the area A and the coodinate (x',y') of the elemental area

dA can be posititve, negative and zero, the product of inertia Ixy can therefore also be posititve, negative and

zero.

Example of Product Moment of Inertia of a Right Angle Triangle

Product Moment of Inertia of a Right Angle Triangle by Double Integration

The product moment of an area A of a right angle triangle about the axes xy is:

Product Moment of Inertia of a Right Angle Triangle by Parallel-axis Theorem

Assuming the elemental area is a rectangular area, the product moment of an elemental area dA of a right

angle triangle about the centroidal axes x'y' is:

The product moment of an elemental area dA of a right angle triangle about the centroidal axis xy is:

Principal Axes and Principal Moments of Inertia

Transformation of Moments of Inertia

In general, the rectangular moments and product of inertia about the rectangular coordinate system at the pole

O is depending on the orientation of the rectangular reference axes of which the rectangular moments and

product of intertia with respect to. Consider an area A is located in a plane with a system of rectangular

coordinate, x and y through the pole O, the rectangular moments and product of inertia of the area A with

respect to the axes, x and y are:

A new rectangular coordinate axes u and v can be obtained by rotating the rectangular coordinate axes x and

y about the pole O by an angle θ. The moments and product of inertia of the area A with respect to the new

axes, u and v are:

The coordinates of the elemental area on the two rectangular coordinate systems can be related by the rotating

angle θ. Imply:

If Ix, Iy and Ixy are known, then:

Since the polar moment of inertia about the origin is equal to the sum of paired rectangular moments of inertia.

Imply:

Principal Axes of Moments of Inertia

Therefore the sum of paired rectangular moments of intertia about a pole is alway a constant regardless of the

orientation of the coordinate axes. Since the rectangular moment is a funtion of angle θ, a maximum or a

minimum value can be obtained by differentiating either one of the rectangular moments of inertia. Imply:

Through differentiation, the angle θ of the rectangular coordinate axes, at which the rectangular moments of

inertia is either maximum or minimum, can be determined. These paired rectangular coordinate axes is called

principal axes and the paired rectangular moments of inertia is called principal moments of inertia. The origin

O of the rectangular coordinate axes can be located inside or outside the area, if the origin O chosen is coincide

with the centroid, the two principal axes of the area about the axes through its centroid C are called the

principal centroidal axes of the area. Since a tangent function is a periodic function with period π, the equation

defines two values 2θm of 180o apart and the two values of θm are 90o apart which is confirmed with the

maximum and minimum values of rectangular moments of inertia about the rectangular coordinate axes, i.e.

the principal moments of inertia about O. Imply:

Principal Moments of Inertia

Besides, one more relation between the rectangular moments of inertia and the product of inertia at the

principal axes is obtained. The product of inertia with respect to the principal axes can also be determined.

Imply:

Therefore the product of inertia on the principal axes is equal to zero. Since the product of inertia is also equal

to zero if one or both of the rectangular axes is an axis of symmetry of the area and is independent on the

location of the origin O, an axis of symmetry is a principal axis, but a principal axis may not be an axis of

symmetry. Since the product of inertia is equal to zero, the rectangular principal moments of an area can be

expressed as:

Besides the sum of the rectangular moments of inertia Ix and Iy is an invarient of the system. The group terms

of value square of R is also an invarient under the rotation of the rectangular coordinate axes. Imply:

Or in an alternate form. Imply:

The group terms on the left hand side of value the product of rectangular moments of inertia minus the square

of product of inertia is also an invarient under the rotation of the rectangular coordinate axes.

Considerações:

O significado fisico do produto de inércia relaciona-se com a distribuição geométrica segundo os eixos. Se

um ou ambos os eixos são de simetria, o produto de inércia é nulo.

O Produto de Inércia é calculado simultaneamente em relação ao par de eixos de referência.

De acordo com a distribuição da área da figura plana ao redor dos eixos de referência, o Produto de

Inércia poderá resultar em um número positivo, negativo ou nulo.

O Teorema de Steiner também é válido para o Produto de Inércia.

Se, pelo menos um eixo de referência for eixo de simetria, o Produto de Inércia resultará nulo.

O resultado do Produto de Inércia de uma figura composta, em relação a um par de eixos, é igual à

soma dos Produtos de Inércia das Figuras Planas, componentes da figura composta, em relação ao

mesmo par de eixos.

Referência de Sinais para Produtos de Inércia:

1º e 3º quadrantes Ixy > 0

2º e 4º quadrantes Ixy < 0

PRODUTOS DE INÉRCIA DE ALGUMAS FIGURAS BÁSICAS

Figura Área Produto de Inércia

A BH

2 2

0

4

g gx y

xy

I

B HI

2

BHA

2 2

2 2

72

24

g gx y

xy

B HI

B HI

2

4

RA

4

4

0,0165

8

g gx y

xy

I R

RI

2

2

RA

0

0

g gx y

xy

I

I

2A R

0

0

g gx y

xy

I

I