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The concept of the inertia tensor ⇐ ПредыдущаяСтр 5 из 5
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 mi (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 Taking into consideration that rix = xi; riy = yi; riz=zi ; (ri, Vector ratio (3.19) can be written in the form of projections on the coordinate axis. Lx = = = ω x Similarly, the projection of angular momentum on both axes y and z may be gotten Ly = ω y Lz = ω z 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 Jxx, Jyy, Jzz are moments of inertia about relevant axes that are called an axial moments of inertia. Other coefficients of proportionality between L and 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 Jxz and Jyz. Consequently, ratios (3.20) – (3.22) may be written in the form of Thus, vectors Lx=J Thus, vectors Now, suppose that vector Lx=Jxz That is the three components of vector The set of nine quantities that are set by ratios (3.23) and (3.24) is It is the inertia tensor*. The inertia tensor Jzz characterizes inert properties of the body in the process of rotation. Quantities Jxx Jyy and Jzz are diagonal components of tensor and other are not diagonal. If quantities that are disposed of symmetric about diagonal, equal to Jxy = Jyx; Jxz = Jzx; Jyz = Jzy, 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 Jxx:
Jxx =
ρ = ρ (x, y, z) is a density; dv is a volume of mass dm.
Thus, the contribution of the moment of inertia of the column in Jzz is defined by dJzzcolumn = 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 Jxx and Jyy: Now, calculate one of the products of inertia that is according to (3.24) Jxy = 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: Jxy = 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 Jz, Jy, Jz 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) Lx = Ly = 0; Lz = Jz ω. Thus, the vector 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 If the vector 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.
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