Product of Inertia

Product of inertia

The product of inertia of area A relative to the indicated XY rectangular axes is I_XY = ∫ xy dA (see illustration). The product of inertia of the mass contained in volume V relative to the XY axes is I_XY = ∫ xyρ dV—similarly for I_YZ and I_ZX.

Product of inertia of an area

Relative to principal axes of inertia, the product of inertia of a figure is zero. If a figure is mirror symmetrical about a YZ plane, I_ZX = I_XY = 0. See Moment of inertia

Product of Inertia

a quantity that characterizes the mass distribution in a body or mechanical system. Products of inertia are the sums of the products formed by multiplying the mass m_k of each point of the body or system by the product of two of the coordinates x_k, y_k, z_k of the point

I_xy = ∑ m_kx_ky_k

I_yz = ∑ m_ky_kz_k

I_zx = ∑ m_kz_kx_k

The values of the products of inertia depend on the orientations of the coordinate axes. For every point of the body or system, there exist at least three mutually perpendicular axes, called the principal axes of inertia, for which the products of inertia are equal to zero.

The concept of the product of inertia plays an important role in the study of the rotational motion of bodies. The magnitudes of the pressure forces on the bearings on which the axial shaft of a body rotates depend on the values of the products of inertia. The pressures will be minimal—that is, equal to the static pressures—if the axis of rotation is a principal axis of inertia that passes through the center of mass of the body.

product of inertia

[′prä·dəkt əv i′nər·shə] (mechanics) Relative to two rectangular axes, the sum of the products formed by multiplying the mass (or, sometimes, the area) of each element of a figure by the product of the coordinates corresponding to those axes.

单词	product of inertia
释义	Product of Inertia Product of inertia The product of inertia of area A relative to the indicated XY rectangular axes is I_XY = ∫ xy dA (see illustration). The product of inertia of the mass contained in volume V relative to the XY axes is I_XY = ∫ xyρ dV—similarly for I_YZ and I_ZX. Product of inertia of an area Relative to principal axes of inertia, the product of inertia of a figure is zero. If a figure is mirror symmetrical about a YZ plane, I_ZX = I_XY = 0. See Moment of inertia Product of Inertia a quantity that characterizes the mass distribution in a body or mechanical system. Products of inertia are the sums of the products formed by multiplying the mass m_k of each point of the body or system by the product of two of the coordinates x_k, y_k, z_k of the point I_xy = ∑ m_kx_ky_k I_yz = ∑ m_ky_kz_k I_zx = ∑ m_kz_kx_k The values of the products of inertia depend on the orientations of the coordinate axes. For every point of the body or system, there exist at least three mutually perpendicular axes, called the principal axes of inertia, for which the products of inertia are equal to zero. The concept of the product of inertia plays an important role in the study of the rotational motion of bodies. The magnitudes of the pressure forces on the bearings on which the axial shaft of a body rotates depend on the values of the products of inertia. The pressures will be minimal—that is, equal to the static pressures—if the axis of rotation is a principal axis of inertia that passes through the center of mass of the body. product of inertia [′prä·dəkt əv i′nər·shə] (mechanics) Relative to two rectangular axes, the sum of the products formed by multiplying the mass (or, sometimes, the area) of each element of a figure by the product of the coordinates corresponding to those axes.
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