The concept of the inertia tensor

Unless the axis is fastened around which the body rotates, but only its point, the movement of the body becomes complicated. At every moment, such movement rotates about its axis that goes through the attachment point.

1. Let us consider the rigid body that is attached to the center of mass O, where is the zero point of coordinate system x, y, z. The rigid body is considered as a set of material points with the mass m_i (fig 3.8). An angular momentum of any material point about point O is

Let us use the distribution rule of double vector product

And write the ratio (3.17) in the form of

The angular momentum about the point O is

It follows that the angular momentum vector generally does not lie in direction of angular rate vector . We shall find the connection between components of these vectors.

Taking into consideration that

r_ix = x_i; r_iy = y_i; r_iz=z_i ;

(r_i, ) = x_iω _x + y_iω _y + z_iω _z,

Vector ratio (3.19) can be written in the form of projections on the coordinate axis.

L_x = {()ω _x – x_i (x_iω _x + y_iω _y + z_iω _z)} =

= ) ω _x – (y_iω _y + z_iω _z)} = 3.20

= ω _x ) – ω _y ω _z .

Similarly, the projection of angular momentum on both axes y and z may be gotten

L_y = ω _y ω _x (3.21)

L_z = ω _z . (3.22)

Coefficients of projections of angular rate ω _x, ω _y, ω _z are measured in kg · m² and depend on the distribution of mass in the body and instantaneous orientation of the body about x, y, z axes.

Let us introduce symbols:

Where J_xx, J_yy, J_zz are moments of inertia about relevant axes that are called an axial moments of inertia. Other coefficients of proportionality between L and are called products of inertia and they are marked in such a way:

Products of inertia are characteristics of dynamic unbalance of the bodies. For example, in the process of rotation about the z axis, a press force on bearings where the axis is fixed depends on values of J_xz and J_yz. Consequently, ratios (3.20) – (3.22) may be written in the form of

Thus, vectors and are not collinear for the arbitrary-shaped body with an arbitrary distribution of mass. Mutual orientation of vectors and is defined by values of coefficients of proportionality between them. If, for example, J_xx=J_yy=J_zz=J and all products of inertia equals to zero i.e. J_xy=J_yz=J_xz=0, the ratios (3.25) acquire the form

L_x=J _x ; L_y = J _y; L_z= J _z. (3.26)

Thus, vectors and in this case are collinear.

Now, suppose that vector points in the direction of z axis i.e. ≠ 0 and _x= _y= 0. Suppose coefficients J_xz ≠ 0, J_yz≠ 0 and J_zz ≠ 0. Substituting the values in (3.25), it is possible to acquire

L_x=J_{xz x} ≠ 0; L_y = J_{yz y} ≠ 0; L_z = J_{zz z} ≠ 0,

That is the three components of vector do not equal to zero. Consequently, vector with the vector in the direction of axis z form an angle.

The set of nine quantities that are set by ratios (3.23) and (3.24) is

It is the inertia tensor*. The inertia tensor J_zz characterizes inert properties of the body in the process of rotation. Quantities J_xx J_yy and J_zz are diagonal components of tensor and other are not diagonal. If quantities that are disposed of symmetric about diagonal, equal to J_xy = J_yx; J_xz = J_zx; J_yz = J_zy, such tensor is called symmetric. In case of continuous distribution of mass, the addition is changed into integration.

The tensor is an ordered set of nine quantities that are called tensor components. Tensor components depend on the choice of coordinate system and in the case of rotation of axes they are changed into product of components of two vectors.

For example, there is a component J_xx:

J_xx = (xyz)(y²+z²)dv.

ρ = ρ (x, y, z) is a density; dv is a volume of mass dm.

As an example, calculate tensor components of inertia for homogeneous rectangular parallelepiped that has such dimensions 2a× 2b× 2c (fig. 3.9). Suppose that the zero point lies in the center of inertia C and axes are paralleled with edges of parallelepiped through the centers of which they go. Calculate the moment of inertia J_zz = Then divide the body into columns which have the square at the basis that equal to dxdy. All components of such column have the same values of x and y axes. The volume of the column is dv = 2cdxdy and it mass is dm = ρ 2cdxdy.

Thus, the contribution of the moment of inertia of the column in J_zz is defined by

dJ_zzcolumn = ρ 2cdxdy(x² + y²).

On integrating this expression into x, we shall find the contribution that the layer which has the length 2a, the height 2c and the weigh dy:

Finally, on integrating into y we shall get the moment of inertia for the entire body:

The relevant calculations give values of J_xx and J_yy:

Now, calculate one of the products of inertia that is according to (3.24)

J_xy = .

The contribution of the moment of inertia of the column with the basis dzdy is

And the contribution of the layer is

Accordingly, the moment of inertia of the entire body equals to zero: J_xy = 0. For other products of inertia, the result is the same. Thus, in the case of a relevant choice of axes, the inertia tensor is significantly simplified and in the result it has only three diagonal components:

Thus, the tensor (3.28) is diagonalized. Dimensions J_z, J_y, J_z are called principal moments of inertia and exes are called principal axes of inertia. The principal axis of inertia crosses over at the center of mass and its directions may be defined by the general consideration of symmetry: in the case of cylinder these are axis of cylinder and perpendicular to it two mutually perpendicular axes, and three mutually perpendicular axes are for the sphere.

Suppose that the body turns around one of its principal axes of inertia, for example around its z axis: ω = ω _z; ω _x = ω _y = 0. Consequently, we have according to (3.25)

L_x = L_y = 0; L_z = J_z ω.

Thus, the vector lies in the direction with .

The same results are obtained when the body turns around other principal axes. At any case the result is

,

Where J is the principal moment of inertia.

Consequently, the character of rotation of rigid body depends on the arrangement of axes of rotation about principal tensor axes of inertia.

If the axis of rotation lies in the principal tensor axes of inertia, the angular momentum lies in the direction of the angular rate vector . When a body moves, the axis of rotation coincides with the direction of the angular momentum. Thus, the axis of rotation retains its rotation direction. In addition, all centrifugal forces are balanced and if the body is fixed on such axis, the additional force that appears during its rotation does not influence on resistance. Thus, such axes are called free.

If the vector lies in no principal axes of inertia, it does not also lie in fixed in space vector ; so, the direction of axis of rotation constantly is changed.

The study of dynamic stability with J₁ > J₂> J₃ demonstrate that the rotation about the free axis for which the moment of inertia J has the maximum value J = J₁ is the most stable. The rotation about the axis with minimum value J= J₃ is constant in the case of absence of exciting force.