Changing moments of inertia with parallel transfer of axes. Changing the moments of inertia of the shear with parallel moving of the axes
Changing the moments of inertia of the shear at parallel transfer axes.
In addition to the static moments, we look at three more advanced integrals:
Previously, through x and y, the current coordinates of the elementary area dF are known in a sufficiently taken coordinate system xOy. The first 2 integrals are called axial moments of inertia the choice of x and y axes is clear. The third integral is called center moment of inertia overcut well x, y. Axis moments are always positive, because area dF is considered positive. The central moment of inertia can be both positive and negative, it can be stale in terms of expansion along the length of the x, y axes.
We will show the formula for the transformation of the moment in inertia with parallel transfer of the axes. (Div pic). Importantly, we need to set the moments of inertia and static moments for the x 1 and y 1 axes. It is necessary to calculate the moments of the x2 and y2 axes.
Substituting here x 2 \u003d x 1 -a and y 2 \u003d y 1 -b Known
Crooked bows, maybe.
If the axis x 1 and y 1 are central, then S x 1 = S y 1 = 0 and otrimani virazi say:
When the axes are moved in parallel (for example, one of the axes is central), the axial moments of inertia change by an amount that increases the area of the cross section by a square between the axes.
2. Static moments of the area across the width of the axes Ozі Oy(div 3, m 3):
4. Central moment of inertia across the width of the axes Ozі Ouch(div 4, m 4):
Oscilki, then
Axis Jzі Jy that polar J p moments of inertia are always positive, shards under the sign of the integral are coordinates of another world. Static moments Szі Sy, as well as the central moment of inertia Jzy can be both positive and negative.
In the range of rolled steel for coils, the values of the center moments behind the module are indicated. The rozrahunkas have the following to acquire their meanings for the improvement of the sign.
For the designation of the sign of the center point of the coil (Fig. 3.2), it is noticeable that it looks like the sum of three integrals, which are counted only for the parts of the periphery, which are spread at the quarters of the coordinate system. It is obvious that for the parts, spreading in the 1st and 3rd quarters, we will have a positive value of the integral, zydA will be positive, and the integrals that are calculated for the parts, spreading in the II and IV quarters will be negative (tvir zydA be negative). Otzhe, for the kutochka in fig. 3.2, and the value of the central moment of inertia will be negative.
Rozmirkovuyuchi similar rank for recutting, so that if you want one whole symmetry (Fig. 3.2, b) you can make a visnovka, so the central moment of inertia J zy is equal to zero, because one of the axes (Oz or Oy) is completely symmetrical to the cut. Definitely, for the parts of tricot, roztashovannyh in 1 and 2 quarters of the water center, the moments of inertia are taken out only by a sign. It can be said that there are several parts that are found in III and IV quarters.
Static moments Assigned to the center of importance
Calculable static moments for a wide range of axes Ozі Oy the rectangle shown in Fig. 3.3.
Rice. 3.3. Until the calculation of static moments
Here: BUT- Crossing area, yCі z C- Coordinates of the center of gravity. The center of gravity of the rectangle is changed on the diagonals.
Obviously, if the axis, where static moments are calculated, pass through the center of gravity of the figure, then its coordinates will reach zero ( z C = 0, yC= 0), i, similar to formula (3.6), static moments and equal to zero. in such a manner, the center of gravity of the crossover is the point that can have such power: the static moment, whatever the axis, to pass through it,zero.
Formulas (3.6) make it possible to know the coordinates of the center of gravity z Cі yC recut folding form. Yakshcho peretin can be given at the sight n parts, which are in the area of the center of gravity, then the calculation of the coordinates of the center of gravity of the entire cross section can be written as:
  .
| (3.7) |
Changing moments of inertia with parallel transfer of axes
Let me see moments of inertia Jz, Jyі Jzy shodo axes Oyz. It is necessary to calculate the moment of inertia J Z, J Yі JZY shodo axes O 1 YZ, parallel to the axes Oyz(fig. 3.4) a(horizontal) and b(vertically)
Rice. 3.4. Changing moments of inertia with parallel transfer of axes
Coordinates of the elementary maidanchik dA bind yourself with such equivalences: Z = z + a; Y = y + b.
Let's calculate the moments of inertia J Z, J Yі JZY.
 | (3.8) |
 | (3.9) |
 | (3.10) |
What a point O axes Oyz run with a dot W- the center of gravity of the peresis (Fig. 3.5); static moments Szі Sy become equal to zero, and the formulas say Y i Zi It is necessary to take with the improvement of symbols. On the axis of the moment of inertia, the signs of the coordinates do not fit (the coordinates are moved to another step), and the axis on the central moment of inertia, the sign of the coordinates in the line (creation Z i Y i A i may be negative).
We introduce the Cartesian rectangular coordinate system Oxy. We can look at the area of coordinates of a certain overcut (closed area) from area A (Fig. 1).
Static moments
Point C with coordinates (x C, y C)
called center of gravity.
If the coordinate axes pass through the center of gravity of the edge, then the static moments of the edge will reach zero:
Axial moments of inertia traversing the x and y axes are called integrals of the form:
Polar moment of inertia The intersection of the cob of coordinates is called the integral of the form:
Central moment of inertia the section is called the integral of the mind:
The head axes of inertia are cut are called two mutually perpendicular to the axis, where I xy =0. As for the mutually perpendicular axes є all the symmetry of the cut, then I xy \u003d 0 i, also, qi axis - smut. Head axles that pass through the center of gravity of the cut are called head central axes of inertia
2. The Steiner-Huygens theorem about the parallel transfer of axes
The Steiner-Huygens theorem (the Steiner theorem).
The axial moment of inertia of the cross section I is about a fairly stable axis x is more than the sum of the axial moment of inertia of the cross section of I from the visual parallel axis x * , which passes through the center of the mass cross section, and the additional area of the cross section A is per square of the axis two d.
If we take into account the moments of inertia I x і I y for the x and y axes, then for the axes ν and u, rotated by kut α, the moments of inertia of the axis and the center of gravity are calculated using the formulas:
From pointing the formulas, it is clear that
Tobto. the sum of axial moments of inertia does not change when turning mutually perpendicular axes, so. . Head axles that pass through the center of gravity of the cut are called head central axes pererazu. For symmetrical cross-sections of the axis and symmetry with the head central axes. The position of the head axes of the cross section of the other axes is determined by the vicarious spіvvіdnoshennia:
de? The axes of the moment of inertia, like the head axes, are called head moments of inertia:
the plus sign in front of another addendum is brought up to the maximum moment of inertia, the minus sign - up to the minimum.
Often, in the case of practical tasks, it is necessary to designate the moments of inertia across the axes, differently oriented at the same plane. If you need to manually tweak the value of the moment in the inertia of the entire crossover (above all warehouse parts) there are other axes that can be found in the technical literature, special indicators and tables, and also look after the formulas. Therefore, it is important to establish fallows between the moments of inertia of one and the same crossover of different axes.
In the wild change, the transition from the old to the new coordinate system can be seen as two successive transformations of the old coordinate system:
1) a path of parallel translation of the coordinate axes at the new position
2) a way to turn їх shоdo a new cob of coordinates. Let's look at the first of these transformations, that is, parallel transfer of the coordinate axes.
It is acceptable that the moments of inertia of the thogo cross section of the old axes (Fig. 18.5) are in the house.
Let's take a new coordinate system of axes that are parallel to ourselves. Significantly a and b are the coordinates of the point (that of the new cob of coordinates) in the old coordinate system
Let's take a look at the elementary area Coordinates її y of the old coordinate system equals y i . The new system stinks equally
We can represent the value of the coordinates of the axial moment of inertia around the axis
In a different way - the moment of inertia is the static moment of the crossover along the axis of the road area F of the crossover.
Otzhe,
If everything z passes through the center of gravity of the cut, then the static moment i
From the formula (25.5) it can be seen that the moment of inertia should be like an axis, so as not to pass through the center of gravity, greater than the moment of inertia for the axis that passes through the center of gravity, by the value of the yoke is positive. From the same moment of inertia for parallel axes, the axial moment of inertia may least value how to pass through the center of gravity of the cut.
Moment of inertia about axis [by analogy to formula (24.5)]
In an okremy fall, if everything passes through the center of gravity of the cut
Formulas (25.5) and (27.5) are widely used when calculating the axial moments of inertia of folding (warehouse) overruns.
Now we can imagine the value of the central moment of inertia for the width of the axes
If the axis is central, then the axis of the moment should look: 
15.Fallow land moments of inertia when turning axes:
J x 1 \u003d 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;
Kut a>0, which means that the transition from the old coordinate system to the new one takes a year. J y 1 + J x 1 = J y + J x
Extreme (maximum and minimum) values of the moment of inertia are called head moments of inertia. Axes, where such axes moments of inertia may have extreme values, are called head axes of inertia. The main axes of inertia are mutually perpendicular. Vіdtsentrovі moments of inertia shоdo main axes = 0, then. The main axes of inertia are the axes, where any water center moment of inertia = 0. As one of the axes, the offenses are escaping from the axis of symmetry, all the stinks are smut. Kut, which determines the position of the main axes: 
so a 0 >0 Þ the axes turn in the opposite direction. All the maximum should be set to a smaller kut z tієї osі, so that the moment of inertia can be more significant. Head axles that pass through the center of the vaga are called head central axes of inertia. Moments of inertia for these axes: 
Jmax + Jmin = Jx + Jy. The central moment of inertia is equal to the head central axes of inertia equal to 0. As a result, the head moment of inertia, the formula for the transition to the rotated axes:
J x 1 \u003d J max cos 2 a + J min sin 2 a; J y 1 \u003d J max cos 2 a + J min sin 2 a; J x 1 y1 = (J max - J min) sin2a;
Kіntsevoi method of calculation of geometrical indications in resection and designation of the main central moments of inertia and the position of the main central axes of inertia. 
Radius of inertia - 
; J x = F x i x 2 , J y = F x i y 2 .
If J x ta J y head moments of inertia, then i x ta i y - head radii of inertia. Elips, promptings on the head radii of inertia, like on the pivos, are called ellipse of inertia. For the help of the elіps of inertia, you can graphically know the radius of inertia i x 1 for any axis x 1. For this you need to draw a dot to ellipse, parallel to the x-axis 1 and decrease the distance from the center of the axis to dot. Knowing the radius of inertia, it is possible to calculate the moment of inertia of the cut along the axis x 1: . For perepіzіv, scho can have more than two axes of symmetry (for example: colo, square, ring and іn) axis moments of inertia along all central axes are equal to each other, J xy \u003d 0, elіps іnertsiy roll up to the stake of inertia.
Enthusiasm...