Axes of inertia. Principal axes and principal moments of inertia Principal axes of the section

The main, three mutually perpendicular axes drawn through the c.-l. point of the body and possessing the property that if they are taken as coordinate axes, then the centrifugal moments of inertia of the body about these axes will be equal to zero. If tv. a body fixed at one point is brought into rotation around an axis, which at a given point is yavl. main O. and., then the body in the absence of external forces will continue to rotate around this axis, as if around a fixed one. The concept of the main O. and. plays an important role in TV dynamics. body.

Physical Encyclopedic Dictionary. - M.: Soviet Encyclopedia..1983 .

AXIS OF INERTIA

The main ones are three mutually perpendicular axes drawn through Ph.D. point of the body, coinciding with the axes of the ellipsoid of inertia of the body at this point. Main O. and. have the property that if they are taken as coordinate axes, then the centrifugal moments of inertia of the body about these axes will be equal to zero. If one of the coordinate axes, e.g. axis Oh, is for point ABOUT the main O. and., then the centrifugal moments of inertia, the indices of which include the name of this axis, i.e. Ixy And I xz, are equal to zero. If a solid body, fixed at one point, is brought into rotation around an axis, the edge at a given point is the main O. and., then the body in the absence of external. forces will continue to rotate around this axis, as around a fixed one.

Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia.Editor-in-Chief A. M. Prokhorov.1988 .

Axial moments of inertia of the section about the axes X And at(see fig. 32, A) are called definite integrals of the form

When determining the axial moments of inertia, in some cases one has to meet with another new geometric characteristic of the section - the centrifugal moment of inertia.

centrifugal moment of inertia sections about two mutually perpendicular axes x y(see fig. 32, A)

Polar moment of inertia section relative to the origin ABOUT(see fig. 32, A) is called a definite integral of the form

Where R- distance from the origin of coordinates to the elementary area d.A.

The axial and polar moments of inertia are always positive, and the centrifugal moment, depending on the choice of axes, can be positive, negative, or equal to zero. Units of designation of moments of inertia - cm 4, mm 4.

There is the following relationship between the polar and axial moments of inertia:

According to formula (41), the sum of the axial moments of inertia about two mutually perpendicular axes is equal to the polar moment of inertia about the point of intersection of these axes (the origin).

Moments of inertia of sections about parallel axes, some of which are central (x s, us)> are determined from the expressions:

Where and iv- coordinates of the center of gravity C of the section (Fig. 34).

Formulas (42), which have great practical application, are read as follows: the moment of inertia of a section about any axis is equal to the moment of inertia about an axis parallel to it and passing through the center of gravity of the section, plus the product of the cross-sectional area by the square of the distance between the axes.

note: coordinates a and c should be substituted into formulas (42) given above, taking into account their signs.

Rice. 34.

From formulas (42) it follows that of all the moments of inertia about parallel axes, the smallest moment will be about the axis passing through the center of gravity of the section, i.e., the central moment of inertia.

The formulas for determining the strength and rigidity of a structure include moments of inertia, which are calculated relative to axes that are not only central, but also principal. In order to determine which axes passing through the center of gravity are the main ones, one must be able to determine the moments of inertia about the axes rotated relative to each other at a certain angle.

Dependences between the moments of inertia during the rotation of the coordinate axes (Fig. 35) have the following form:

Where A- angle of rotation of the axes And And v about the axes henna respectively. Angle a is considered positive if the rotation of the axes And and u going counterclock-wise.

Rice. 35.

The sum of the axial moments of inertia about any mutually perpendicular axes does not change when they are rotated:

When the axes rotate around the origin, the centrifugal moment of inertia changes continuously, therefore, at a certain position of the axes, it becomes equal to zero.

Two mutually perpendicular axes with respect to which the centrifugal moment of inertia of the section is equal to zero are called principal axes of inertia.

The direction of the principal axes of inertia can be determined as follows:

Two values ​​of the angle obtained from formula (43) A differ from each other by 90° and give the position of the principal axes. As you can see, the smaller of these angles does not exceed in absolute value l /4. In the future, we will use only a smaller angle. The main axis drawn at this angle will be denoted by the letter And. On fig. 36 shows some examples of the designation of the main axes in accordance with this rule. Initial axes are denoted by letters hee w.

Rice. 36.

In bending problems, it is important to know the axial moments of inertia of sections relative to those principal axes that pass through the center of gravity of the section.

Principal axes passing through the center of gravity of the section are called main central axes. In what follows, as a rule, for brevity we will call these axes simply main axes, omitting the word "central".

The axis of symmetry of a plane section is the main central axis of inertia of this section, the second axis is perpendicular to it. In other words, the axis of symmetry and any one perpendicular to it form a system of principal axes.

If a plane section has at least two axes of symmetry that are not perpendicular to each other, then all axes passing through the center of gravity of such a section are its main central axes of inertia. So, in fig. 37 shows some types of sections (circle, ring, square, regular hexagon, etc.), which have the following property: any axis passing through their center of gravity is the main one.

Rice. 37.

It should be noted that non-central principal axes are of no interest to us.

In the theory of bending, the moments of inertia about the main central axes are of greatest importance.

Principal central moments of inertia or main moments of inertia are the moments of inertia about the main central axes. Moreover, relative to one of the main axes, the moment of inertia maximum, relative to the other - minimal:

Axial moments of inertia of the sections shown in fig. 37, calculated relative to the main central axes, are equal to each other: J y , Then: J u = J x cos 2 a + J y sin a = J x .

The moments of inertia of a complex section are equal to the sum of the moments of inertia of its parts. Therefore, to determine the moments of inertia of a complex section, we can write:

where eJ xi , J y „ J xiyi - moments of inertia of individual parts of the section.

NB: if the section has a hole, then it is convenient to consider it as a section with a negative area.

To perform strength calculations in the future, we introduce a new geometric characteristic of the strength of a bar working in a straight bend. This geometric characteristic is called the axial moment of resistance or the moment of resistance in bending.

The ratio of the moment of inertia of a section about an axis to the distance from this axis to the most distant point of the section is called axial moment of resistance:

The moment of resistance has the dimension of mm 3, cm 3.

Moments of inertia and moments of resistance of the most common simple sections are determined by the formulas given in Table. 3.

For rolled steel beams (I-beams, channel beams, angle beams, etc.), the moments of inertia and moments of resistance are given in the tables of the assortment of rolled steels, where, in addition to the dimensions, the cross-sectional areas, the positions of the centers of gravity and other characteristics are given.

In conclusion, we introduce the concept radius of gyration sections relative to the coordinate axes X And at - i x And i y respectively, which are determined by the following formulas.

Main axes - these are the axes with respect to which the axial moments of inertia take extreme values: minimum and maximum.

Principal central moments of inertia are calculated with respect to principal axes passing through the center of gravity.

Examples of problem solving

Example 1 Determine the value of the axial moments of inertia of a flat figure about the axes Oh And OU(Fig. 25.5).

Solution

1. Determine the axial moment of inertia about the axis Oh. We use formulas for principal central moments. We represent the moment of inertia of the section as the difference between the moments of inertia of a circle and a rectangle.

For a circle

For rectangle

For rectangle axis Oh does not pass through the CT. Moment of inertia of a rectangle about an axis Oh:

where A is the cross-sectional area; a - distance between axles Oh And Oh oh.

Moment of inertia of section

Example 2 Find the main central moment of inertia of the section about the axis Oh(Fig. 25.6).

Solution

1. The section is made up of standard profiles, the main central moments of inertia of which are given in the GOST tables, see Appendix 1. For an I-beam No. 14 according to GOST 8239-89 Jox 1 \u003d 572 cm 4.

For channel No. 16 according to GOST 8240-89 Jox 2 \u003d 757 cm 4.

Area A 2 \u003d 18.1 cm 2, Jo y 2 = 63.3 cm 4 .

2. Determine the coordinate of the center of gravity of the channel relative to the axis Oh. In a given section, the channel is turned and raised. In this case, the main central axes have changed places.

y 2 \u003d (h 1 / 2) + d2-zo 2 , according to GOST we find h 1 = 14 cm; d2= 5 mm; z o = 1.8 cm.

The moment of inertia of the section is equal to the sum of the moments of inertia of the channels and the I-beam about the axis Oh. We use the formula for the moments of inertia about parallel axes:

In this case

Example 3 For a given section (Fig. 2.45), calculate the main central moments of inertia.

Solution

The section has two axes of symmetry, which are its main central axes.

We divide the section into two simple shapes: a rectangle ( I) and two circles (II).

Moment of inertia of the section about the axis X

Axis x(the central axis of the section) is not the central axis of the circle. Therefore, the moment of inertia of the circle should be calculated by the formula

Substituting values J x '' , a, F" into the formula, we get

Axis at is the center of the rectangle and circles. Hence,

Example 4 For a given section (Fig. 2.46), determine the position of the main central axes and calculate the main central moments of inertia.

Solution

The center of gravity lies on the Oy axis, since it is the axis of symmetry of the section. Dividing the section into two rectangles I(160 x 100) and II(140 x 80) and selecting the minor axis and, determine the coordinate of the center of gravity v0 according to the formula

axes Oh And OU- main central axes of the section ( OU- axis of symmetry, axis Oh passes through the center of gravity of the section and is perpendicular to OU).

Calculate the main moments of inertia of the section J x And Jy:

The y-axis is the central axis for rectangles 1 And 11. Hence,

To check the correctness of the solution, you can divide the section into rectangles in another way and re-calculate. The coincidence of the results will confirm their correctness.

Example 5 Calculate the main central moments of inertia of the section (Fig. 2.47).

Solution

The section has two axes of symmetry, which are its main central axes.

Divide the section into two rectangles with b*h= 140 x 8 and two rolled channels. For channel No. 16 from the table GOST 8240 - 72 we have J X 1 = J x = 747 cm 4; J y 1 \u003d 63, 3 cm 9, F1\u003d 18.1 cm 2, z0= 1.8cm.

Calculate J x and J y:

Example 6 Determine the position of the main central axes and calculate the main central moments of inertia of a given section (Fig. 2.48).

Solution

We divide the given section into rolled profiles: channel I and two I-beams II. We take the geometric characteristics of the channel and I-beam from the tables of rolled steel GOST 8240-72 and GOST 8239 - 72.

For channel No. 20 JXl = 113 cm 4 (in the table Jy); Jy 1 \u003d 1520 cm 4 (in the table Jx); F1= 23.4 cm 2; G 0 = 2.07 cm.

For I-beam No. 18 J x 2= 1330 cm 4 (in the table J x); Jy 2 = 94.6 cm 4 (in table J y); F 2 = 23.8 cm2.

One of the main axes is the axis of symmetry OU, another major axis Oh passes through the center of gravity of the section perpendicular to the first.

Choosing a secondary axis And and determine the coordinate v0:

Where v1= 180 + 20.7 = 200.7 mm and v2= 180/2 = 90 mm. Calculate J x And J at:

Control questions and tasks

1. The diameter of the solid shaft was increased by 2 times. How many times will the axial moments of inertia increase?

2. The axial moments of the section are equal, respectively J x = 2.5 mm 4 and J y = 6.5mm. Determine the polar moment of the section.

3. Axial moment of inertia of the ring about the axis Oh J x = 4 cm 4 . Determine the value J p .

4. In which case J x the smallest (Fig. 25.7)?

5. Which of the following formulas for determining J x suitable for the section shown in Fig. 25.8?

6. Moment of inertia of channel No. 10 relative to the main central axis JXQ= 174cm 4; cross-sectional area 10.9 cm 2 .

Determine the axial moment of inertia about the axis passing through the base of the channel (Fig. 25.9).

7. Compare the polar moments of inertia of two sections that have almost the same area (Fig. 25.10).

8. Compare axial moments of inertia about the axis Oh rectangle and square having the same area (Fig. 25.11).

Principal axes of inertia and principal moments of inertia.

When the angle changes, the values ​​of Ix1, Iy1 and Ix1y1 change. Find the value of the angle at which Ix1 and Iy1 have extreme values; to do this, we take the first derivative with respect to Ix1 or Iy1 and set it equal to zero: or from where (1.28)

This formula determines the position of two axes, relative to one of which the axial moment of inertia is maximum, and relative to the other it is minimal.

Such axes are called principal. The moments of inertia about the principal axes are called principal moments of inertia.

We find the values ​​of the main moments of inertia from formulas (1.23) and (1.24), substituting into them from formula (1.28), while using the known trigonometry formulas for the functions of double angles.

After transformations, we obtain the following formula for determining the main moments of inertia: (1.29)

By examining the second derivative, it can be established that for this case (Ix< Iy) максимальный момент инерции Imax имеет место относительно главной оси, повернутой на угол по отношению к оси х, а минимальный момент инерции - относительно другой, перпендикулярной оси. В большинстве случаев в этом исследовании нет надобности, так как по конфигурации сечений видно, какая из главных осей соответствует максимуму момента инерции.

The main axes passing through the center of gravity of the section are called the main central axes.

In many cases, it is possible to immediately determine the position of the main central axes. If the figure has an axis of symmetry, then it is one of the main central axes, the second passes through the center of gravity of the section perpendicular to the first. The foregoing follows from the fact that with respect to the axis of symmetry and any axis perpendicular to it, the centrifugal moment of inertia is equal to zero.

If the two main central moments of inertia of the section are equal to each other, then any central axis of this section is the main one, and all the main central moments of inertia are the same (circle, square, hexagon, equilateral hexagon).

9. Basic geometric characteristics of sections

Here: C- center of gravity of flat sections;

A- cross-sectional area;

I x , I y- axial moments of inertia of the section relative to the main axes;

I xI , I yI- axial moments of inertia relative to the auxiliary axes;

I p- polar moment of inertia of the section;

W x , W y- axial moments of resistance;

W p- polar moment of resistance

Rectangular section

Cross section of an isosceles triangle

10. The main types of forces acting on the body. Moment of force about the center. Moment properties.

When considering mechanical problems, most of the forces acting on bodies can be attributed to three main varieties:

The force of gravity;

Friction force;

Elastic force.

All the bodies around us are attracted to the Earth, this is due to the action of the forces of universal gravitation. If we neglect air resistance, then we already know that all bodies fall to the Earth with the same acceleration - the acceleration of free fall.

Like any object, a body suspended on a spring tends to fall down due to the gravity of the Earth, but when the spring stretches to a certain length, the body stops, that is, it comes to a state of mechanical equilibrium. We already know that mechanical equilibrium occurs when the sum of the forces acting on the body is zero. This means that the force of gravity acting on the load must balance with some force acting from the side of the spring. This force, directed against the force of gravity and acting from the side of the spring, is called the elastic force.

After passing a certain distance, the body stops, the speed of the body decreases from the initial value to zero, that is, the acceleration of the body is a negative value. Consequently, a force acts on the body from the side of the surface, which tends to stop this body, that is, it acts against its speed. This force is called friction force.

Moment of force about the center (point).

Moment of force F relative to the center (point) ABOUT called vector m o (F) equal vector product vector radius r drawn from the center ABOUT exactly A force application, on the force vector F:

where the shoulder h is a perpendicular dropped from the center ABOUT to the line of force F.

Moment m o (F) characterizes the rotational effect of the force F about the center (point) ABOUT.

Properties of the moment of force:

1. Moment of force about the center does not change when transferring power along the line of action to any point;

2. If line of action force passes through the center ABOUT(h = 0), then the moment of force about the center ABOUT zero.

The axes about which the centrifugal moment of inertia is zero are called principal, and the moments of inertia about these axes are called principal moments of inertia.

Let us rewrite formula (2.18) taking into account the known trigonometric relations:

;

in this form

In order to determine the position of the main central axes, we differentiate equality (2.21) with respect to the angle α once we obtain

At a certain value of the angle α=α 0 , the centrifugal moment of inertia may turn out to be zero. Therefore, taking into account the derivative ( V), the axial moment of inertia will take on an extreme value. Equating

,

we obtain a formula for determining the position of the main axes of inertia in the form:

(2.22)

In formula (2.21), we take cos2 out of brackets α 0 and substitute the value (2.22) there and, taking into account the well-known trigonometric dependence we get:

After simplification, we finally obtain a formula for determining the values ​​of the main moments of inertia:

(2.23)

Formula (20.1) is used to determine the moments of inertia about the principal axes. Formula (2.22) does not give a direct answer to the question about which axis the moment of inertia will be maximum or minimum. By analogy with the theory of studying a plane stress state, we present more convenient formulas for determining the position of the principal axes of inertia:

(2.24)

Here α 1 and α 2 determine the position of the axes, relative to which the moments of inertia are respectively equal J 1 and J 2. In this case, it should be borne in mind that the sum of the modules of the angles α 01 and α 02 must equal π/2:

Condition (2.24) is the orthogonality condition for the principal axes of inertia of a plane section.

It should be noted that when using formulas (2.22) and (2.24) to determine the position of the main axes of inertia, the following pattern must be observed:

The main axis, relative to which the moment of inertia is maximum, makes the smallest angle with the original axis, relative to which the moment of inertia is greater.

Example 2.2.

Determine the geometric characteristics of flat sections of the beam relative to the main central axes:

Solution

The proposed section is asymmetric. Therefore, the position of the central axes will be determined by two coordinates, the main central axes will be rotated relative to the central axes at a certain angle. This implies such an algorithm for solving the problem of determining the main geometric characteristics.

1. We divide the section into two rectangles with such areas and moments of inertia about their own central axes:

F 1 \u003d 12 cm 2, F 2 \u003d 18 cm 2;

2. We set the system of auxiliary axes X 0 at 0 starting at point A. The coordinates of the centers of gravity of the rectangles in this system of axes are as follows:

X 1 =4 cm; X 2 =1 cm; at 1 \u003d 1.5 cm; at 2 \u003d 4.5 cm.

3. Determine the coordinates of the center of gravity of the section according to the formulas (2.4):

We apply the central axes (in red in Fig. 2.9).

4. We calculate the axial and centrifugal moments of inertia about the central axes X with and at c according to formulas (2.13) as applied to a composite section:

5. We find the main moments of inertia according to the formula (2.23)

6. Determine the position of the main central axes of inertia X And at by formula (2.24):

The main central axes are shown in (Fig. 2.9) in blue.

7. Let's check the performed calculations. To do this, we will carry out the following calculations:

The sum of the axial moments of inertia about the main central and central axes must be the same:

The sum of the modules of the angles α X and α y,, defining the position of the main central axes:

In addition, the provision that the main central axis X, relative to which the moment of inertia J x has a maximum value, makes a smaller angle with that central axis, relative to which the moment of inertia is greater, i.e. with axle X With.