AXIS OF INERTIA
AXIS OF INERTIA
The main, three mutually perpendicular axes drawn through the k.-l. point of the body and having the property that if they are taken as coordinate axes, then the centrifugal inertia of the body relative to these axes will be equal to zero. If TV a body fixed at one point is put into rotation around an axis, which at a given point is manifested. main O. and., then the body in the absence of external. forces will continue to rotate around this axis, as if around a stationary one. The concept of the main O. and. plays important role
in the TV speaker. bodies. Physical encyclopedic Dictionary. . 1983 .
AXIS OF INERTIA
. - M.: Soviet Encyclopedia The main ones are three mutually perpendicular axes drawn through the k.n. 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 relative to these axes will be equal to zero. If one of the coordinate axes, for example. axis Oh, is for the point ABOUT main O. and., centrifugal moments of inertia, the indices of which include the name of the axis, i.e. I xy And I xz
, are equal to zero. If a solid body, fixed at one point, is brought into rotation around an axis, which at a given point is the main object of rotation, then the body, in the absence of external forces will continue to rotate around this axis, as if around a stationary one.. Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. 1988 .
Editor-in-Chief A. M. Prokhorov
See what "AXIS OF INERTIA" is in other dictionaries: The main three mutually perpendicular axes, which can be drawn through any point of a solid body, differ in that if a body fixed at this point is brought into rotation around one of them, then in the absence of external forces it will... ...
Big Encyclopedic Dictionary Main, three mutually perpendicular axes that can be drawn through any point of a solid body, characterized in that if a body fixed at this point is brought into rotation around one of them, then in the absence of external forces it will... ...
The main, three mutually perpendicular axes drawn through some point of the body, having the property that, if they are taken as coordinate axes, then the centrifugal moments of inertia (See Moment of inertia) of the body relative to these axes ... ... Great Soviet Encyclopedia
The main, three mutually perpendicular axes, which can be drawn through any point on the TV. bodies, characterized in that if a body fixed at this point is brought into rotation around one of them, then in the absence of external it will continue... Natural science. encyclopedic Dictionary
main axes of inertia- Three mutually perpendicular axes drawn through the center of gravity of the body, having the property that if they are taken as coordinate axes, then the centrifugal moments of inertia of the body relative to these axes will be equal to zero.... ... Technical Translator's Guide
main axes of inertia- three mutually perpendicular axes drawn through the center of gravity of the body, having the property that if they are taken as coordinate axes, then the centrifugal moments of inertia of the body relative to these axes will be equal to zero.... ...
- ... Wikipedia
Main axes- : See also: main axes of inertia, main axes (tensor) of deformation... Encyclopedic Dictionary of Metallurgy
Dimension L2M SI units kg m² SGS ... Wikipedia
Moment of inertia scalar physical quantity, characterizing the distribution of masses in the body, equal to the sum products of elementary masses by the square of their distances to the base set (point, line or plane). SI unit: kg m².… … Wikipedia
Books
- Thoretic physics. Part 3. Solid mechanics (2nd edition), A.A. Eichenwald. The third part of this course in theoretical physics is a natural continuation of part II: the basic principles of mechanics are applied here to solid body, i.e. to the system...
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 and 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 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 put cos2 out of brackets α
0 and substitute the value (2.22) there and, taking into account the known trigonometric dependence 
we get:
After simplification, we finally obtain the 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 main 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 for studying a plane stress state, we present more convenient formulas for determining the position of the main axes of inertia:
(2.24)
Here α 1 and α 2 determine the position of the axes about which the moments of inertia are respectively equal J 1 and J 2. It should be borne in mind that the sum of the angle modules α 01 and α 02 should equal π/2:
Condition (2.24) is the condition for the orthogonality of the main 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.
Define geometric characteristics flat sections of timber relative to the main central axes:
 |
Solution
The proposed section is asymmetrical. 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 by a certain angle. This leads to an algorithm for solving the problem of determining the main geometric characteristics.
1. We divide the section into two rectangles with the following areas and moments of inertia relative to their own central axes:
F 1 =12 cm 2, F 2 =18 cm 2;
2. We define a system of auxiliary axes X 0 at 0 starting at point A. The coordinates of the centers of gravity of the rectangles in this axis system are as follows:
X 1 =4 cm; X 2 =1 cm; at 1 =1.5 cm; at 2 =4.5 cm.
3. Determine the coordinates of the center of gravity of the section using formulas (2.4):
We plot the central axes (in red in Fig. 2.9).
4. Calculate the axial and centrifugal moments of inertia relative to the central axes X with and at c according to formulas (2.13) as applied to the composite section:
5. Find the main moments of inertia using formula (2.23)
6. Determine the position of the main central axes of inertia X I xy at according to formula (2.24):
The main central axes are shown in (Fig. 2.9) in blue.
7. Let's check the calculations performed. 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:
Sum of angle modules α X and α y,, defining the position of the main central axes:
In addition, the provision is fulfilled that the main central axis X, about which the moment of inertia J x has the maximum value, makes a smaller angle with the central axis relative to which the moment of inertia is greater, i.e. with axle X With.
The main, three mutually perpendicular axes drawn through the k.-l. point of the body and having the property that if they are taken as coordinate axes, then the centrifugal moments of inertia of the body relative to these axes will be equal to zero. If TV a body fixed at one point is put into rotation around an axis, which at a given point is manifested. main O. and., then the body, in the absence of external forces, will continue to rotate around this axis, as if around a stationary one. The concept of the main O. and. plays an important role in the dynamics of TV. bodies.
Physical encyclopedic dictionary. - M.: Soviet Encyclopedia..1983 .
AXIS OF INERTIA
. - M.: Soviet Encyclopedia The main ones are three mutually perpendicular axes drawn through the k.n. 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 relative to these axes will be equal to zero. If one of the coordinate axes, for example. axis Oh, is for the point ABOUT main O. and., centrifugal moments of inertia, the indices of which include the name of the axis, i.e. I xy And I xz
, are equal to zero. If a solid body, fixed at one point, is brought into rotation around an axis, which at a given point is the main object of rotation, then the body, in the absence of external forces will continue to rotate around this axis, as if around a stationary one..Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia.1988 .
Task 5.3.1: For the section, the axial moments of inertia of the section relative to the axes are known x1, y1, x2: , . Axial moment of inertia about the axis y2 equal...
1) 1000 cm4; 2) 2000 cm4; 3) 2500 cm4; 4) 3000 cm4.
Solution: The correct answer is 3). The sum of the axial moments of inertia of the section relative to two mutually perpendicular axes when the axes are rotated through a certain angle remains constant, that is
After substituting the given values, we get:
Task 5.3.2: Of the indicated central axes of the section of an equal angle angle, the main ones are...
1) x3; 2) everything; 3) x1; 4) x2.
Solution: The correct answer is 4). For symmetrical sections, the axes of symmetry are the main axes of inertia.
Task 5.3.3: Main axes of inertia...
- 1) can only be drawn through points lying on the axis of symmetry;
- 2) can only be drawn through the center of gravity of a flat figure;
- 3) these are the axes about which the moments of inertia of a flat figure are equal to zero;
- 4) can be drawn through any point of a flat figure.
Solution: The correct answer is 4). The figure shows an arbitrary flat figure. Through the point WITH two mutually perpendicular axes are drawn U I xy V.
In the course on strength of materials it is proven that if these axes are rotated, then their position can be determined in which the centrifugal moment of inertia of the area becomes zero, and the moments of inertia about these axes take extreme values. Such axes are called main axes.
Task 5.3.4: Of the indicated central axes, the main section axes are...
1) everything; 2) x1 I xy x3; 3) x2 And x3; 4)x2 I xy x4.
Solution: The correct answer is 1). For symmetrical sections, the axes of symmetry are the main axes of inertia.
Task 5.3.5: Axes about which the centrifugal moment of inertia is zero and the axial moments take extreme values are called...
- 1) central axes; 2) axes of symmetry;
- 3) main central axes; 4) main axes.
Solution: The correct answer is 4). When the coordinate axes are rotated by an angle b, the moments of inertia of the section change.
Let the moments of inertia of the section relative to the coordinate axes be given x, y. Then the moments of inertia of the section in the system of coordinate axes u, v, rotated at a certain angle relative to the axes x, y, are equal
At a certain value of the angle, the centrifugal moment of inertia of the section becomes zero, and the axial moments of inertia take extreme values. These axes are called main axes.
Task 5.3.6: Moment of inertia of the section about the main central axis xC equal...
1); 2) ; 3) ; 4) .
Solution: The correct answer is 2)
To calculate we use the formula
Principal axes of inertia and principal moments of inertia.
When the angle changes, the quantities Ix1, Iy1 and Ix1y1 change. Let us find the value of the angle at which Ix1 and Iy1 have extreme values; to do this, take the first derivative of Ix1 or Iy1 with respect to and equalize it 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 - minimal.
Such axes are called main axes. 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), and we use the well-known trigonometry formulas for 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 we can establish that for this case (Ix< Iy) максимальный момент инерции Imax имеет место относительно main axis, rotated at an angle relative to the x-axis, and the minimum moment of inertia is relative to another, perpendicular to the axis. In most cases, this study is not necessary, since the configuration of the sections shows which of the main axes corresponds to the maximum moment of inertia.
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 a figure has an axis of symmetry, then it is one of the main central axes, the second passing through the center of gravity of the section perpendicular to the first. The above follows from the fact that relative 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 a section are equal to each other, then for this section any central axis 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
Section of an isosceles triangle
10. The main types of forces acting on the body. Moment of force about the center. Properties of moment of forces.
When considering mechanical problems, most forces acting on bodies can be classified into three main types:
The force of universal gravity;
Friction force;
Elastic force.
All bodies around us are attracted to the Earth, this is due to the action of the forces of universal gravity. If we neglect air resistance, then we already know that all bodies fall to the Earth with the same acceleration - the acceleration of gravity.
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 exerted by the spring. This force, directed against gravity and acting from the spring, is called elastic force.
After traveling 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 surface, which tends to stop this body, that is, acts against its speed. This force is called friction force.
Moment of force relative to the center (point).
A moment of power F relative to the center (point) ABOUT called a vector m o (F) equal vector product radius of the vector r carried out from the center ABOUT exactly A application of force to the force vector F:
where arm h is a perpendicular dropped from the center ABOUT to the line of action of force F.
Moment m o (F) characterizes the rotational effect of force F relative to the center (point) ABOUT.
Properties of moment of force:
1. Moment of force relative to the center does not change when transferring force along the line of its action to any point;
2. If line of action strength passes through the center ABOUT(h = 0), then the moment of force relative to the center ABOUT equal to zero.
