Stress calculation of moment transmitting roll with profile on the base of Reuleaux triangle |
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19/06/2014 9:30am
Detailed calculation of polar moment of inertia for cross section of Reuleaux triangle form is considered. With the help of received dependences, it is possible to determine to a high precision concerning stress, appearing in profile connection during torque transmission. Also software module for strength prediction of profiling rolls is described. It allows to fulfill strength check and adjustment of overall parameters of profile momenttransmitting connection on the base of Reuleaux triangle. UDK 531.8 Razumov Mikhail Sergeyevich
South-West State University, Kursk, Russia Stress calculation of moment transmitting roll with profile on the base of Reuleaux triangleTorsion analysis of moment transmitting connections lies mainly in determination of maximum tangential stress in cross section. Basic calculating formula for determination of tangential stress in the point y , where Тmax – maximum torque, transmitted by the roll, (1) where I xc and Iyc - axial moments of inertia.
Figure 1 Determination of polar moment of inertia of Reuleaux triangle Triangle side c (parameter R in the fig. 1) may be determined from the formula . Altitude of the triangle h may be determined as follows Triangle polar moment of inertia is equal to (2) Polar moment of inertia of circular segment may be determined from the formula (fig. 2): (3) where I X1 segm and I Y1 segm - axial moments of segment;
Now let us determine axial moment of inertia I X1 segm, I Y1 segm and circular segment area (fig. 2, b). Figure 2 b Determination of polar moment of inertia of circular segment about the axis YC. As one knows from mechanics of solid deformable body: and Calculating I X1 segm, surface element is located at a distance y = R•sinα. Correspondingly dy = R•cosα dα. Value b = R•cosα. Area of surface element equals dA = 2b•dy. The angle may change within. Circular segment area is equal to (4) Moment of inertia concerning X C1 axis equals (5) Let us calculate I Yc segm where x= Rcos t - distance from Y axis to surface element. Correspondingly, dx = -Rsin tdt . Consequently, limits on integral will be changed: at x1=0, , at , . As a result we will get (6) First moment of area of segment in x-axis equals Coordinate of gravitational center may be determined from the formula
(7)
Finally, inserting (4), (5), (6) and (7) into (3), we will get polar moment of inertia of circular segment: (8) Let us assume polar moment of inertia of Reuleaux triangle in the form of the sum oftriangle polar moment of inertia and tripled polar moment of inertia of circular segment: (9) On the base of these calculations, computer program was developed. Width across corners and maximum torque, being transmitted by profile connection, are the input data. Program interface is shown in the figure 3. Figure 3 Program interface a) calculation route selection b) calculation in accordance with allowable carrier power c) selection of minimum size of cross section The given above mathematical tool may be reprocessed and realized for calculation of five-cornered connection. However Reuleaux pentagon is impractical among machine-building profiles. Research article is fulfilled with funding from RF President’s grant for government support of young Russian scholars - Doctors of Philosophy MK - 2653.2014.8. References
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