Moment of inertia during parallel translation of axes. Changing moments of inertia during parallel translation of coordinate axes Formulas for translating axes

The axes passing through the center of gravity of a plane figure are called central axes.
The moment of inertia about the central axis is called the central moment of inertia.

Theorem

The moment of inertia about any axis is equal to the sum of the moment of inertia about the central axis parallel to the given one and the product of the area of ​​the figure and the square of the distance between the axes.

To prove this theorem, consider an arbitrary plane figure whose area is equal to A , the center of gravity is located at the point WITH , and the central moment of inertia about the axis x will Ix .
Let's calculate the moment of inertia of the figure relative to a certain axis x 1 , parallel to the central axis and spaced from it at a distance A (rice).

I x1 = Σ y 1 2 dA + Σ (y + a) 2 dA =
= Σ y 2 dA + 2a Σ y dA + a 2 Σ dA.

Analyzing the resulting formula, we note that the first term is the axial moment of inertia relative to the central axis, the second term is the static moment of the area of ​​this figure relative to the central axis (hence, it is equal to zero), and the third term after integration can be represented as a product a 2 A , i.e., as a result we get the formula:

I x1 = I x + a 2 A- the theorem is proven.

Based on the theorem, we can conclude that of a series of parallel axes, the axial moment of inertia of a flat figure will be the smallest relative to the central axis .

Principal axes and principal moments of inertia

Let us imagine a flat figure whose moments of inertia relative to the coordinate axes Ix And I y , and the polar moment of inertia relative to the origin is equal to I ρ . As was established earlier,

I x + I y = I ρ.

If the coordinate axes are rotated in their plane around the origin of coordinates, then the polar moment of inertia will remain unchanged, and the axial moments will change, while their sum will remain constant. Since the sum of variables is constant, one of them decreases and the other increases, and vice versa.
Consequently, at a certain position of the axes, one of the axial moments will reach the maximum value, and the other - the minimum.

Axes about which the moments of inertia have a minimum and maximum value, are called the main axes of inertia.
The moment of inertia about the main axis is called the main moment of inertia.

If the principal axis passes through the center of gravity of a figure, it is called the principal central axis, and the moment of inertia about such an axis is called the principal central moment of inertia.
We can conclude that if a figure is symmetrical about any axis, then this axis will always be one of the main central axes of inertia of this figure.

Centrifugal moment of inertia

The centrifugal moment of inertia of a flat figure is the sum of the products of elementary areas taken over the entire area and the distance to two mutually perpendicular axes:

I xy = Σ xy dA,

Where x , y - distances from the site dA to axles x And y .
The centrifugal moment of inertia can be positive, negative or zero.

The centrifugal moment of inertia is included in the formulas for determining the position of the main axes of asymmetrical sections.
Standard profile tables contain a characteristic called radius of gyration of the section , calculated by the formulas:

i x = √ (I x / A),i y = √ (I y / A) , (hereinafter the sign"√"- root sign)

Where I x , I y - axial moments of inertia of the section relative to the central axes; A - cross-sectional area.
This geometric characteristic is used in the study of eccentric tension or compression, as well as longitudinal bending.

Torsional deformation

Basic concepts about torsion. Torsion of a round beam.

Torsion is a type of deformation in which only a torque occurs in any cross section of the beam, i.e. a force factor that causes a circular movement of the section relative to an axis perpendicular to this section, or prevents such movement. In other words, torsional deformations occur if a pair or pairs of forces are applied to a straight beam in planes perpendicular to its axis.
The moments of these pairs of forces are called twisting or rotating. Torque is denoted by T .
This definition conventionally divides the force factors of torsional deformation into external ones (torsional, torque T ) and internal (torques M cr ).

In machines and mechanisms, round or tubular shafts are most often subjected to torsion, so strength and rigidity calculations are most often made for such units and parts.

Consider the torsion of a circular cylindrical shaft.
Imagine a rubber cylindrical shaft in which one of the ends is rigidly fixed, and on the surface there is a grid of longitudinal lines and transverse circles. We will apply a couple of forces to the free end of the shaft, perpendicular to the axis of this shaft, i.e. we will twist it along the axis. If you carefully examine the grid lines on the surface of the shaft, you will notice that:
- the shaft axis, which is called the torsion axis, will remain straight;
- the diameters of the circles will remain the same, and the distance between adjacent circles will not change;
- longitudinal lines on the shaft will turn into helical lines.

From this we can conclude that when a round cylindrical beam (shaft) is torsioned, the hypothesis of flat sections is valid, and we can also assume that the radii of the circles remain straight during deformation (since their diameters have not changed). And since there are no longitudinal forces in the shaft sections, the distance between them is maintained.

Consequently, the torsional deformation of a round shaft consists in the rotation of the cross sections relative to each other around the torsion axis, and their rotation angles are directly proportional to the distances from the fixed section - the farther any section is from the fixed end of the shaft, the greater the angle relative to the shaft axis it twists .
For each section of the shaft, the angle of rotation is equal to the angle of twist of the part of the shaft enclosed between this section and the seal (fixed end).


Corner ( rice. 1) rotation of the free end of the shaft (end section) is called the full angle of twist of the cylindrical beam (shaft).
Relative twist angle φ 0 called the torsion angle ratio φ 1 to the distance l 1 from a given section to the embedment (fixed section).
If the cylindrical beam (shaft) is long l has a constant cross-section and is loaded with a torsional moment at the free end (i.e., consists of a homogeneous geometric section), then the following statement is true:
φ 0 = φ 1 / l 1 = φ / l = const - the value is constant.

If we consider thin layer on the surface of the above rubber cylindrical bar ( rice. 1), limited by a grid cell cdef , then we note that this cell warps during deformation, and its side, remote from the fixed section, shifts towards the twist of the beam, occupying the position cde 1 f 1 .

It should be noted that a similar picture is observed during shear deformation, only in this case the surface is deformed due to translational movement of sections relative to each other, and not due to rotational movement, as in torsional deformation. Based on this, we can conclude that during torsion in cross sections, only tangential internal forces (stresses) arise, forming a torque.

So, the torque is the resulting moment relative to the axis of the beam of internal tangential forces acting in the cross section.

If the axes are central, then the moment axes will look like:

15.Dependency between moments of inertia when turning the axes:

J x 1 =J x cos 2 a + J y sin 2 a - J xy sin2a; J y 1 =J y cos 2 a + J x sin 2 a + J xy sin2a;

J x 1 y1 = (J x - J y)sin2a + J xy cos2a ;

Angle a>0, if the transition from the old coordinate system to the new one occurs counterclockwise. J y 1 + J x 1 = J y + J x

Extreme (maximum and minimum) values ​​of moments of inertia are called main moments of inertia. The axes about which the axial moments of inertia have extreme values ​​are called main axes of inertia. The main axes of inertia are mutually perpendicular. Centrifugal moments of inertia about the main axes = 0, i.e. main axes of inertia - axes about which the centrifugal moment of inertia = 0. If one of the axes coincides or both coincide with the axis of symmetry, then they are the main ones. Angle defining the position of the main axes: , if a 0 >0 Þ the axes rotate counterclockwise. The maximum axis always makes a smaller angle with that of the axes relative to which the moment of inertia has a greater value. The main axes passing through the center of gravity are called main central axes of inertia. Moments of inertia about these axes:

J max + J min = J x + J y . The centrifugal moment of inertia relative to the main central axes of inertia is equal to 0. If the main moments of inertia are known, then the formulas for transition to rotated axes are:

J x 1 =J max cos 2 a + J min sin 2 a; J y 1 =J max cos 2 a + J min sin 2 a; J x 1 y1 = (J max - J min)sin2a;

The ultimate goal of calculating the geometric characteristics of the section is to determine the main central moments of inertia and the position of the main central axes of inertia. Radius of inertia - ; J x =F×i x 2 , J y =F×i y 2 .

If J x and J y are the main moments of inertia, then i x and i y - principal radii of inertia. An ellipse built on the main radii of inertia as on the semi-axes is called ellipse of inertia. Using the ellipse of inertia, you can graphically find the radius of inertia i x 1 for any axis x 1. To do this, you need to draw a tangent to the ellipse, parallel to the x1 axis, and measure the distance from this axis to the tangent. Knowing the radius of inertia, you can find the moment of inertia of the section relative to the x axis 1: . For sections with more than two axes of symmetry (for example: circle, square, ring, etc.), the axial moments of inertia about all central axes are equal, J xy =0, the ellipse of inertia turns into a circle of inertia.

Given: moments of inertia of the figure relative to the z, y axes; distances between these and parallel axes z 1, y 1 – a, b.

Determine: moments of inertia about the z 1, y 1 axes (Fig. 4.7).

Coordinates of any point in new system z 1 Oy 1 can be expressed in terms of coordinates in the old system like this:

z 1 = z + b, y 1 = y + a.

We substitute these values ​​into formulas (4.6) and (4.8) and integrate term by term:

In accordance with formulas (4.1) and (4.6), we obtain

,

, (4.13)

If the initial data of the zCy axis are central, then the static moments S z and

S y are equal to zero and formulas (4.13) are simplified:

,

, (4.14)

.

Example: determine the axial moment of inertia of the rectangle relative to the z 1 axis passing through the base (Fig. 4.6, a). According to formula (4.14)

4.4. Dependence between moments of inertia when turning axes

Given: moments of inertia of an arbitrary figure relative to the coordinate axes z, y; the angle of rotation of these axes α (Fig. 4.8). We consider the counterclockwise rotation angle to be positive.

Determine: moments of inertia of the figure relative to z 1, y 1.

The coordinates of an arbitrary elementary area dF in the new axes are expressed through the coordinates of the previous system of axes as follows:

z 1 = OB = OE + EB = OE + DC = zcos α + ysin α,

y 1 = AB = AC – BC = AC – ED = ycos α – zsin α.

Let's substitute these values ​​into (4.6) and (4.8) and integrate term by term:

,

,

Taking into account formulas (4.6) and (4.8), we finally find:

. (4.16)

Adding formulas (4.15), we get: (4.17)

Thus, when the axes rotate, the sum of the axial moments of inertia remains constant. In this case, each of them changes in accordance with formulas (4.15). It is clear that at some position of the axes the moments of inertia will have extreme values: one of them will be the largest, the other the smallest.

4.5. Principal axes and principal moments of inertia

The main central axes, the centrifugal moment of inertia about which is zero, are of greatest practical importance. We will denote such axes by the letters u, υ. Consequently, J uυ = 0. The initial arbitrary coordinate system z, y must be rotated by such an angle α 0 that the centrifugal moment of inertia becomes equal to zero. Equating (4.16) to zero, we obtain

. (4.18)

It turns out that the theory of moments of inertia and the theory of plane stress state are described by the same mathematical apparatus, since formulas (4.15) – (4.18) are identical to formulas (3.10), (3.11) and (3.18). Only instead of normal stresses σ axial moments of inertia J z and J y are recorded, and instead of tangential stresses τ zy - centrifugal moment of inertia J zy. Therefore, we present the formulas for the main axial moments of inertia without derivation, by analogy with formulas (3.18):

.(4.19)

The two values ​​of the angle α 0 obtained from (4.18) differ from each other by 90 0 , the smaller of these angles does not exceed 45 0 in absolute value.

Radius of inertia and moment of resistance

The moment of inertia of a figure relative to any axis can be represented as the product of the area of ​​the figure by the square of a certain quantity, called radius of gyration:

, (4.20)

where i z is the radius of gyration relative to the z axis.

From expression (4.20) it follows that

,
. (4.21)

The main central axes of inertia correspond to the main radii of inertia

,
. (4.22)

Knowing the main radii of inertia, you can graphically find the radius of inertia (and, consequently, the moment of inertia) relative to an arbitrary axis.

Let's consider another geometric characteristic that characterizes the strength of the rod during torsion and bending - moment of resistance. The moment of resistance is equal to the moment of inertia divided by the distance from the axis (or from the pole) to the most distant point of the section. The dimension of the moment of resistance is a unit of length cubed (cm 3).

For a rectangle (Fig. 4.6, a)
,
, therefore the axial moments of resistance

,
. (4.23)

For a circle
(Fig. 4.6, b),
, therefore the polar moment of resistance

. (4.24)

For a circle
,
, therefore the axial moment of resistance

. (4.25)

Let us introduce a Cartesian rectangular coordinate system O xy . Let us consider an arbitrary section in the coordinate plane ( closed area) with area A (Fig. 1).

Static moments

Point C with coordinates (x C , y C)

called center of gravity of the section.

If the coordinate axes pass through the center of gravity of the section, then the static moments of the section are equal to zero:

Axial moments of inertia sections relative to the x and y axes are called integrals of the form:

Polar moment of inertia section with respect to the origin of coordinates is called an integral of the form:

Centrifugal moment of inertia section is called an integral of the form:

The main axes of inertia of the section two mutually perpendicular axes are called, relative to which I xy = 0. If one of the mutually perpendicular axes is the axis of symmetry of the section, then I xy =0 and, therefore, these axes are the main ones. The main axes passing through the center of gravity of the section are called main central axes of inertia of the section

2. Steiner-Huygens theorem on parallel translation of axes

Steiner-Huygens theorem (Steiner's theorem).
The axial moment of inertia of section I relative to an arbitrary fixed axis x is equal to the sum of the axial moment of inertia of this section I with the relative axis x * parallel to it, passing through the center of mass of the section, and the product of the cross-sectional area A by the square of the distance d between the two axes.

If the moments of inertia I x and I y relative to the x and y axes are known, then relative to the ν and u axes rotated by an angle α, the axial and centrifugal moments of inertia are calculated using the formulas:

From the above formulas it is clear that

Those. the sum of axial moments of inertia when rotating mutually perpendicular axes does not change, i.e. the axes u and v, relative to which the centrifugal moment of inertia of the section is zero, and the axial moments of inertia I u and I v have extreme values ​​max or min, are called the main axes of the section . The main axes passing through the center of gravity of the section are called main central axes of the section. For symmetrical sections, their axes of symmetry are always the main central axes. The position of the main axes of the section relative to other axes is determined using the relationship:

where α 0 is the angle by which the x and y axes must be rotated so that they become the main ones (a positive angle is usually set counterclockwise, a negative angle is set clockwise). Axial moments of inertia about the principal axes are called main moments of inertia:

The plus sign in front of the second term refers to the maximum moment of inertia, the minus sign to the minimum.

Change in the moments of inertia of the rod at parallel transfer axes.

In addition to static moments, consider the following three integrals:

Where, as before, x and y denote the current coordinates of the elementary area dF in an arbitrary coordinate system xOy. The first 2 integrals are called axial moments of inertia of the section relative to the x and y axes, respectively. The third integral is called centrifugal moment of inertia of the section relative to x, y. Axial moments are always positive, because The area dF is considered positive. The centrifugal moment of inertia can be either positive or negative, depending on the location of the section relative to the x, y axes.

Let us derive formulas for converting moments of inertia during parallel translation of axes. (see pic). We will assume that we are given moments of inertia and static moments about the x 1 and y 1 axes. It is required to determine the moments about the x2 and y2 axes.

Substituting here x 2 =x 1 -a and y 2 =y 1 -b We find

Opening the brackets, we have.

If the x 1 and y 1 axes are central, then S x 1 = S y 1 =0 and the resulting expressions are simplified:

When the axes are transferred in parallel (if one of the axes is the central one), the axial moments of inertia change by an amount equal to the product of the cross-sectional area and the square of the distance between the axes.