## Calculation of shafts for strength and fatigue resistance

## General

The load mode of the shaft is set by the schedule of load changes in time.

The calculation schemes of the shafts and axes of the reducers are in the form of stepped or smooth beams on hinged bearings.

Bearings that simultaneously absorb axial and radial loads are replaced by hinged-fixed supports, and bearings that receive only radial forces are pivotally supported by supports.

The position of the hinged support is determined taking into account the angle of contact a of the rolling bearing (Table A.155-P.160).

For a = 0 for radial bearings, the bearing position is taken in the middle of the bearing width. If the loads acting on the shaft and brought to the axis of the shaft are located in different planes, then they should be decomposed into components lying in two mutually perpendicular planes, and in each of these planes determine the reference reactions and internal forces.

The components of the support reactions and internal forces are summed geometrically.

The loads transferred to the shafts and axes from the part placed on them lead to the center of the connection in the form of concentrated torque T, axial radial forces I x , I y and moments M x , M y> acting in two mutually

perpendicular planes (Figure 5.20).

To bring forces to the geometric axis of the shaft, the distributed load in the gearing is replaced by the concentrated force applied in the middle of the ring gear.

** Fig. 5.20 **

Determination of the concentrated force in the meshing of gears and its projections is discussed in Sec. 3.

The formulas for determining the values of axial and polar resistance moments for different cross sections of shafts are given in Table. 11.184.

## Technique for plotting bending and torque moments

Let's consider a technique of construction of bending and twisting moments, by means of which the values of bending and twisting moments in any section of the shaft for which they are constructed are determined.

The bending moment in the cross-section of the shaft (or axis) is the moment of internal forces applied in this section, which is numerically equal to the algebraic sum of the moments of all external forces acting on one side of the section considered.

The following rule for signs for bending moments is adopted.

The bending moment in the cross section of the beam is considered positive if the resultant moment of external forces to the left of the cross section is directed clockwise, and to the right of the cross section - counterclockwise. He bends the tank bulge down. The bending moment is considered negative if it bends the beam with a bulge upward.

Before the bending moments are plotted, the shaft is divided into sections in the axial direction, on which it is obvious that the magnitude of the moment varies monotonically according to a well-defined law (most often this is a direct proportionality to the length of the section) and determine the values of the moments at the boundaries of this section. According to the obtained data, a graph is constructed of the change in the magnitude of the bending moment along the length of the shaft, which is usually called the bending moment diagram.

The plotting of the diagram is based on the calculation scheme shown under the sketch of the shaft drawing (for better visibility), representing the axis of the shaft, represented as a straight line 1-1.2 mm thick and the length equal to the length of the shaft to which all forces acting on the shaft (both external and reactive) at the same distances from each other and from the ends of the axis as on the shaft and at the same distances from the axis as from the axis of the shaft.

It should be borne in mind here that transverse forces (forces normal to the axis of the shaft) as sliding vectors can be brought to the axis of the shaft.

When constructing diagrams, special attention should be paid to the following:

1. The equations of moments required in the construction of the diagram are compiled with respect to the cross section under consideration on the basis of the force factors acting on one side of the given section.

2. If there are concentrated moments on the shaft (for example, with the action of axial forces applied at a certain distance from the longitudinal axis of the shaft), an instantaneous change in the magnitude of the moment by the value of the concentrated moment, the so-called jump, appears. This jump can be either positive or negative, depending on the sign of the concentrated moment.

3. The bending moment diagrams are constructed in two mutually perpendicular planes. In determining the magnitude of the total bending moment in any section, the above diagrams determine their components and are summed by the Pythagorean theorem.

In this case, it should be borne in mind that in cases where the diagram is located on both sides of the zero line in the section under consideration, a large magnitude of the moment, counted from the zero line (Figures 5.21, 5.22) is taken into account.

For the dangerous section (Figure 5.21), the calculated value of the bending moment is:

The value of M, determined by the formula (5.7), is assumed to be positive.

4. In order for the values M x and A/, to be convenient to summarize, their diagrams are recommended to be built on the same scale.

The torque in the section of the shaft (or axis) is the moment from the tangents of the internal forces, which is numerically equal to the algebraic sum of the external torque acting on one side of the cross section.

To construct the torque curves (a graph showing the change in torque along the shaft length), the generally accepted rule of signs does not exist. For definiteness, it is recommended to consider its value as positive if the resulting external moment applied to the considered part of the shaft will tend to rotate it clockwise (if we look at the cut off part from the section side), and negative - otherwise.

The construction of the torque diagram is obviously derived from the definition of torque.

For the visual representation of the nature of the change in the bending moment along the length of the beam and for finding dangerous sections, diagrams M are constructed. The technique of constructing this time diagram will be considered in the following example.

A drawing of the shaft with its associated parts is drawn in the scale (Figure 5.22).

On the basis of the drawing, a diagram of the shaft with all (active and reactive) forces acting on it is drawn.

Forces in gearing gears are determined on the basis of the calculations given in Sec. 3.2. In this problem, the forces acting from the side of the worm wheel to the worm are determined from the materials of Section. 3.2.3.2.3 (Table 3.34), and the value of the component of the force acting on the belt transmission support, according to the materials of Sec. 3.2.5.3 (Table 3.43).

** Fig. 5.21 **

Distances a. B, c are determined constructively, taking into account the features of the calculation of radial-thrust bearings, set out in Sec. 4.4.2.4.7.

The calculation is then made in the following order.

##### ** 5.Z.6.2.1. Determination of the components of reactions in supports **

The force picture will be considered separately in the planes (Figure 5.22):

• vertical (yOy);

• horizontal (xOg).

###### ** 5.3.6.2.1.1. Plane UOX **

1. Let us compose the equation of the moments acting on the shaft with respect to the support A, starting from the equilibrium condition of the system.

There is no single rule for signs when solving such problems. Here and further, in solving such problems, let us, for definiteness, consider positive the moment acting in the clockwise direction, and negative - acting counter-clockwise.

** **

** Fig. 5.22 **

Then

whence

2. Starting from the equilibrium condition of the system, we will compose the equation of the sum of the forces acting on the shaft:

whence

whence

###### ** 5.Z.6.2.1.2. Plane X02 **

1. Let us compose the equation of the moments acting on the shaft with respect to the support A, starting from the equilibrium condition of the system:

whence

2. Starting from the equilibrium condition of the system, we will compose the equation of the sum of the forces acting on the shaft:

whence

##### ** 5.Z.6.2.2. Construction of bending moment diagrams **

Define the bending moments in the shaft sections that are numerically equal to the algebraic sum of the bending moments of all external forces acting on one side of the cross section.

###### ** 5.Z.6.2.2.1. Plane VOg **

Based on the obtained data, the diagram M x is constructed.

###### ** 5.Z.6.2.2.2. Plane X02 **

Based on the received data, a diagram of m.

##### ** 5.Z.6.2.Z. Construction of torque curves **

Define the torques in the shaft sections that are numerically equal to the algebraic sum of the torques acting on one side of the cross section.

For r 3 = 0 T, s = 0; when 2 ^ - a, T3 = 0.

Based on the obtained data, a diagram of T is constructed.

For more details on plotting the twisting and bending moments, see, for example, in the section "Resistance of materials in the course" Mechanics ".

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