Shaft Design and Analysis



Shaft Design and Analysis

A shaft is the component of a mechanical device that transmits rotational motion and power. It is integral to any mechanical system in which power is transmitted from a prime mover, such as an electric motor or an engine, to other rotating parts of the system. There are many examples of mechanical systems incorporating rotating elements that transmit power: gear-type speed reducers, belt or chain drives, conveyors, pumps, fans, agitators, household appliances, lawn maintenance equipment, parts of a car, power tools, machines around an office or workplace and many types of automation equipment.

Visualize the forces, torques, and bending moments that are created in the shaft during operation. In the process of transmitting power at a given rotational speed, the shaft is inherently subjected to a torsional moment, or torque. Thus, torsional shear stress is developed in the shaft. Also, a shaft usually carries power-transmitting components, such as gears, belt sheaves, or chain sprockets, which exert forces on the shaft in the transverse direction (perpendicular to its axis). These transverse forces cause bending moments to be developed in the shaft, requiring analysis of the stress due to bending. In fact, most shafts must be analyzed for combined stress.

Because of the simultaneous occurrence of torsional shear stresses and normal stresses due to bending, the stress analysis of a shaft virtually always involves the use of a combined stress approach. The recommended approach for shaft design and analysis is the distortion energy theory of failure. Vertical shear stresses and direct normal stresses due to axial loads also occur at times, but they typically have such a small effect that they can be neglected. On very short shafts or on portions of shafts where no bending or torsion occurs, such stresses may be dominant.

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Procedure for Design and analysis of a Shaft

1. Determine the rotational speed of the shaft, n (rpm).

2. Select the material from which the shaft will be made, and specify ultimate tensile strength Su, yield strength Sy and its surface condition: ground, machined, hot-rolled and as-forged. At the moment, due to lack of database for endurance strength, this module should be used in the design and analysis of steel shafts only. Use the database in selection of a material.

3. Apply a desired reliability for definition of reliability factor, CR.

4. Apply a design factor, N (we prefer to use ηd).

5. Propose the general form of the geometry for the shaft, considering how each element on the shaft will be held in position axially and how power transmission from each element to the shaft is to take place. Design details such as fillet radii, shoulder heights, and key-seat dimensions must also be specified. Sometimes the size and the tolerance for a shaft diameter are dictated by the element to be mounted there. For example, ball bearing manufacturers' catalogs give recommended limits for bearing seat diameters on shafts.

6. Specify the location of bearings to support the shaft. The reactions on bearings supporting radial loads are assumed to act at the midpoint of the bearings. Another important concept is that normally two and only two bearings are used to support a shaft. They should be placed on either side of the power-transmitting elements if possible to provide stable support for the shaft and to produce reasonably well-balanced loading of the bearings. The bearings should be placed close to the power-transmitting elements to minimize bending moments. Also, the overall length of the shaft should be kept small to keep deflections at reasonable levels.

7. Determine the design of the power-transmitting components or other devices that will be mounted on the shaft, and specify the required location of each device.

8. Determine the power to be transmitted by the shaft.

9. Determine the magnitude of torque at point of the shaft where the power-transmitting element is.

T = 30 H/π n [N-m]

where:

H = transmitted power, W

T = torque, N-m.

n = rotational speed, rpm.

10. Determine the forces exerted on the shaft.

Spur and helical gears, tangential force

Wt = 60 000 H / π d n [N]

where: d = pitch diameter of gear in [mm];

H = Power in [W];

N = Rotational Speed in [rev/min]

Radial force ; Wr = Wt. tan φn / cos ψ [N]

where: [pic] = normal pressure angle for helical gears, and pressure angle for spur gears; and

ψ = helix angle

11. Preparing a torque diagram.

12. Resolve the radial forces into components in perpendicular directions, vertically and horizontally.

13. Solve for the reactions on all support bearings in each plane.

14. Produce the complete shearing force and bending moment diagrams to determine the distribution of bending moments in the shaft.

15. Analyze each critical point of the shaft to determine the minimum acceptable diameter of the shaft at that point in order to ensure safety under the loading at that point. In general, the critical points are several and include those where a change of diameter takes place, where higher values of torque and bending moment occur, and where stress concentrations occur.

If a vertical shearing force V is the only significant loading present, this equation should be used to compute the required diameter for a shaft.

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where:

Kt = stress concentration factor at the shoulder; 1.5 to 2.5;

V = Vertical Shear Force [N];

N = Factor of Safety / Design Factor (you may use ηd);

D or d = Diameter of the Shaft at the section considered [mm];

S’n = modified endurance strength [MPa], (Which depends on ultimate tensile strength Su).

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where:

Cs = size factor;

CR = reliability factor;

Sn = endurance strength [MPa]

In most shafts, the resulting diameter will be much smaller than that required at other parts of the shaft where significant values of torque and bending moment occur. Also, practical considerations may require that the shaft be somewhat larger than the computed minimum to accommodate a reasonable bearing at the place where the shearing force V is equal to the radial load on the bearing.

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Most shafts are subjected to bending and torsion. The power being transmitted causes the torsion, and the transverse and radial forces on the elements cause bending. In the general case, the transverse forces do not all act in the same plane. In such cases, the bending moment diagrams for two perpendicular planes are prepared first. Then the resultant bending moment at each point of interest is determined.

A design equation is now developed based on the assumption that the bending stress in the shaft is repeated and reversed as the shaft rotates, but that the torsional shear stress is nearly uniform.

[pic]

where:

M = Bending moment (a resultant obtained from bending moment diagrams; (this creates reversed bending stresses on the shaft) [N-mm];

T = Torsion or twisting moment (usually steady) [N-mm];

N = Factor of safety; (We shall usually use η.)

D = Diameter of the shaft at the section under investigation; in [mm].

Also, Sy and Sn are to be taken as [MPa]

Fillet radius

When a change in diameter occurs in a shaft to create a shoulder against which to locate a machine element, a stress concentration dependent on the ratio of the two diameters and on the radius in the fillet is produced. It is recommended that the fillet radius r be as large as possible to minimize the stress concentration, but at times the design of the gear, bearing, or other element affects the radius that can be used. For the purpose of design, we will classify fillets into two categories: sharp and well-rounded.

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The term sharp here does not mean truly sharp, without any fillet radius at all. Such a shoulder configuration would have a very high stress concentration factor and should be avoided. Instead, sharp describes a shoulder with a relatively small fillet radius. One situation in which this is likely to occur is where a ball or roller bearing is to be located. The inner race of the bearing has a factory-produced radius, but it is small. The fillet radius on the shaft must be smaller yet in order for the bearing to be seated properly against the shoulder. When an element with a large chamfer on its bore is located against the shoulder, or when nothing at all seats against the shoulder, the fillet radius can be much larger (well-rounded), and the corresponding stress concentration factor is smaller. We will use the following values for design for bending:

[pic] = 2.5 (sharp fillet)

[pic]= 1.5 (well-rounded fillet)

In this program the factor is determined under the formula (Elastic stress, bending)

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where

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D = larger diameter of the shaft;

d = adjacent smaller diameter of the shaft;

[pic] [pic] source

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Design factor

Under typical industrial conditions, the design factor of N = 3 is recommended. If the application is very smooth, a value as low as N = 2 may be justified. Under conditions of shock or impact, N = 4 or higher should be used, and careful testing is advised.

Desired reliability

This factor is used to apply a reliability factor CR.

Desired reliability Reliability factor

0.50 1.00

0.90 0.90

0.99 0.81

0.999 0.75

Endurance strength data typically reported are average values over many tests, thus implying a reliability of 0.50 (50%). Assuming that the actual failure data follow a normal distribution, the factors from this table can be used to adjust for higher levels of reliability.

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Stress concentration factors

The order of input of the stress concentration factors is “Kt1/Kt2”. First input the value of Kt1. Then enter a slash. Second input the value of Kt2.

For instance: “1.0/3.0”.

Kt1 is the value of the stress concentration factor to the right of a bearing.

Retaining rings are used for many types of locating tasks in shaft applications. The rings are installed in grooves in the shaft after the element to be retained is in place. The ring manufacturer dictates the geometry of the groove. Its usual configuration is a shallow groove with straight side walls and bottom and a small fillet at the base of the groove. The behavior of the shaft in the vicinity of the groove can be approximated by considering two sharp shoulders positioned closed together. Thus, the stress concentration factor for a groove is fairly high.

When bending exists, we will use Kt1 = 3.0 for preliminary design as an estimate to account for the fillets and the reduction in diameter at the groove to determine the nominal shaft diameter before the groove is cut. When torsion exists along with bending, or when only torsion exists at a section of interest, the stress concentration factor is not applied to the torsional shear stress component because it is steady. To account for the decrease in diameter at the groove, however, increase the resulting computed diameter by approximately 6%, a typical value for commercial retaining ring grooves. But after the final shaft diameter and groove geometry are specified, the stress in the groove should be computed with the appropriate stress concentration factor for the groove geometry. The use of a spacing hub (sleeve) for the bearing rests on a shoulder results in a Kt1 =1.0.

Everything, that is said for factor Kt1 concerns and for a factor Kt2.

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Align

This variable can receive one of following values: R – right side align or L – left side align. This factor is used to indicate element position on the shaft.

For example: the value of Align “R” means that parameter “Distance” for the current element is equal to the value of space from beginning of the shaft up to right side of element. The value of Align “L” means that parameter “Distance” for the current element is equal the value of space from beginning of the shaft up to left side of element.

Transmitted power

To understand the method of computing stress in the gear teeth, consider the way power is transmitted by gear system. For discussion, we’ll use the example of a single-reduction gear pair. Power is received from the motor by the input shaft rotating at motor speed. Thus, there is a torque in the shaft can be computed from the following equation:

Torque = power/ rotational speed = H / ω

The input shaft transmits the power from the coupling to the point where the pinion is mounted. The power is transmitted from the shaft to pinion through the key. The teeth of the pinion drive the teeth of the gear thus transmit the power to the gear. But again, power transmission actually involves the application of a torque during rotational at given speed. The torque is the product of the force acting tangent to pitch circle of the pinion times the pitch radius of the pinion. We will use the symbol Wt to indicate the tangential force. As described, Wt is the force exerted by the pinion teeth on the gear teeth. But if the gears are rotating at constant speed and are transmitted a uniform level of power, the system is in equilibrium. Therefore, there must be an equal and opposite tangential force exerted by the gear teeth back on the pinion teeth. This is an application of the principle of action and reaction.

To complete the description of the power flow, the tangential force on the gear teeth produces a torque on the gear equal to the product of Wt times the pitch radius of the gear. Because Wt is the same on the pinion and the gear, but the pitch radius of gear is lager than that of the pinion, the torque on the gear (the output torque) is greater than the input torque. However, note that the power transmitted is the same or slightly less because of mechanical inefficiencies. The power then flows from the through the key to the output shaft and finally to the driven machine.

From this description of power flow, we can see that gears transmit power by exerting a force by the driving teeth on the driving teeth while the reaction force acts back on the teeth of the driving gear. Wt is not the total force on the tooth. Because of the involute form of the tooth, the total force transferred from one tooth to the mating tooth acts normal to the involute profile. For this action we will use symbol Wn. So the tangential force is actually the horizontal component of the total force. Note that there is a vertical component of total force acting radially on the gear tooth Wr .

Note: consumed (received) power should have positive value (>0), and delivered power should have negative value ( ................
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