Elements and Units of Machines
4.1 SELECTING THE BASIC SHAFT SIZE
A shaft is a straight, mostly round bar, which is designed to have other rotating parts of a mechanism mounted on it. To function as such, the shaft may include splines and keyways to transmit torque, journals for rolling or sliding bearings, cylindrical and conical sections, threads, shoulders, and so on. Some elements that are commonly made as separate items (e.g., pinions, cams, connecting flanges) are often made instead as integral parts of the shaft.
Sometimes a shaft is not a part of a mechanism but serves as a connection between mechanisms or machines placed at a distance from each other. Such a shaft, called intermediate, must be provided with elements transmitting torque on its ends; these are, as a rule, flanges, or splines, or necks for couplings.
Although the shape of a shaft depends on the kind and features of the elements adjoined to or mounted on it, the size (or, rather, the diameter) of the shaft is mainly dictated by strength considerations. Not only the strength of the shaft’s body but also the strength of the connections transmitting torque and radial forces, necessary diameters of bearing journals, etc., should also be considered. All these elements should be designed as component parts of a unit. Then, the elements of the shaft should be proportioned so as to avoid large, and/or abrupt differences (changes) in diameter. This may be achieved by selecting a suitable material and heat treatment, changing relations between diameter and length of connections and journal bearings, varying types and dimensions of rolling bearings, changing types of connections (e.g., spline, key, or flange), and by other means. All in all, the process of designing a shaft is an integral part of designing the whole mechanism.
In this iterative process, it is important to make an initial appraisal of the shaft diameter as limited by its strength. The diameter may vary over a wide range depending on environmental conditions, materials used, heat treatment, surface treatment, thoroughness of smoothing out the stress concentrators, and other factors. Tentatively, for the beginning of design, the diameter of a shaft loaded with a torque and bending moments, and with stress raisers such as shoulders, key ways and the like can be estimated from:
(4.1)
where
T= transmitted torque (N·mm)
Su= tensile strength of the shaft material (MPa)
The diameter of a shaft that is loaded with torque only and doesn’t have considerable stress raisers may be as small as one half of the above:
(4.2)
d to T
Su
≈(5 6)3
d to T
Su
≈( .2 5 3)3
44 Machine Elements: Life and Design
EXAMPLE 4.1
A shaft transmits a torque of 15,000 N·m. In addition, it is loaded by bending moments generated by gears and couplings. The tensile strength of the shaft material Su = 930 MPa. The shaft has stress raisers such as press-fit connections, keyways, and shoulders. Limited by fatigue strength, the diameter of the shaft should be approximately equal to
It should be clear that this approximate estimation can’t replace the accepted strength calcula-tion, and the shaft must finally be checked for strength. In some cases, the equations cited earlier are inapplicable, for example, when the torque is about zero (see Figure 6.3 and Figure 6.24). The diameter of each axle in these figures is determined by the inner diameter of the planet wheel bearing and by the bending strength of the axle, which is loaded by the gear teeth forces.
Although decreasing the dimensions of shafts is essential in the effort to limit weight, designers usually don’t use the strongest material for the first design of a new mechanism, and there are three reasons for that. First, such materials are noticeably more expensive and more difficult in machining, and that raises the price. Second, unanticipated loads may be present, which result in breakdowns, forcing a redesign with an increase in shaft strength as the objective. Third, sometimes the customer wants to use the mechanism for greater loads and asks the manufacturer to increase its load rating.
In these occurrences, the increase of the shaft diameter is usually undesirable, because it results in changing the mating parts, often including the housing. This increases by far the upgrade expenses.
If alternatives are available, such as using some stronger material, application of surface hardening (carburizing or nitriding) and cold working, they will save a great deal of expense and effort in these situations.
Sometimes there is no alternative but to increase the shaft diameter to satisfy rigidity or stability requirements, mount larger bearings, or increase the load capacity of connections between the shaft and the attached parts.
EXAMPLE 4.2
Figure 4.20 depicts the main shaft of a subway tunnel escalator. Forces Fcu and Fcl, estimated in dozens of tons, load each of the two hauling sprockets. The diameters of the bearing necks of the shaft have been defined by the bearing’s inner diameter. The diameter of the shaft between the sprockets, as calculated in compliance with the strength requirements, shall be of about 250 mm.
(See dashed lines in Figure 4.20.) But in this case, the elastic bending of the shaft will result in inadmissible tooth misalignment in the reduction gear. To avoid this defect, the shaft diameter has been increased to 450 mm. Calculation of the resilience of this shaft is cited in Section 4.4.
Shafts, which are increased in diameter for the sake of rigidity, can be made of less strong (and consequently cheaper) materials. But even in these cases, the use of soft steels with hardness of HB 140–160 is not a good choice. While assembling tight or press fits, the mounting surfaces may be damaged (i.e., scored or crushed), and the shaft may be bent. An accidental hit with a hard object may leave a remarkable dent on such a shaft. In service, a short-term overload may cause turning of the inner race of a rolling bearing relative to the shaft neck. The soft material of the shaft will be worn away a bit, the bearing fit will become weaker, and the bearing unit will be unable to work (see Chapter 6, Section 6.2).
If the soft shaft is made as an integral part with a flange, the latter may be damaged while tightening bolts (see Chapter 10, Section 10.1). For these and other reasons as well, it is desirable that the shaft be sufficiently hard.
d≈(5to6) 15000 10⋅ = to mm
930 125 150
3 3
Shafts 45
EXAMPLE 4.3
An intermediate steel shaft of 2000 mm length should transmit a constant torque of 2000 Nm without considerable bending loads. Rotational speed is 1000 r/min. The tensile strength of the shaft material Su= 930 MPa. From Equation 4.2, the shaft diameter should be
Taking the shaft diameter value equal to 35 mm, let’s define the natural frequency of its bending vibrations by the following equation1:
(4.3)
where
n= number of the mode shape (n= 1, 2, 3, and so on) L= shaft length between bearings (mm)
E= modulus of elasticity (MPa)
g= 9.81·103 mm/sec2= acceleration of gravity A= shaft cross-sectional area (mm2)
γ= specific weight (density) of the shaft material (N/mm3) I= moment of inertia of the cross section (mm4)
For a solid round shaft made of steel,
Substitution of this data in Equation 4.3 gives us the fundamental frequency that is valid for all solid steel shafts at n= 1:
(4.4)
In our case, d= 35 mm, L= 2000 mm,
The rotational speed is 1000/60 = 16.7 revolutions per second, which is very close to the fundamental frequency f1. This may cause resonance vibrations, and the shaft stiffness should be increased so as to make the f1 value about 1.4 to 1.5 times more than the rotational speed. Inasmuch as the f1 value of a solid shaft is in direct proportion to the shaft diameter (see Equation 4.4), the diameter should be multiplied by 1.4 to 1.5, i.e., to 50 mm roundly. At that diameter, the weight of the shaft doubles,
d≈( .2 5to3) 2000 10⋅ =
46 Machine Elements: Life and Design from 15 kg to 30 kg. If the weight is limited, the shaft should be made hollow, for example, diameters outside (do) and inside (di) may be chosen as do/di= 40/30 mm or 45/38 mm. (These diameters are calculated so as to keep the same section modulus as for the solid shaft of 35 mm in diameter.) Substituting the data of these sections in Equation 4.3 gives us the following values of fundamental frequency:
For section 40/30 mm, f1= 25.1 Hz For section 45/38 mm, f1= 29.6 Hz
Thus, the fundamental frequencies of the hollow shafts are greater than the rotational speed by a factor of 1.55 and 1.77. An additional benefit is that the weight is less (than that of the solid shaft of 35 mm diameter) by a factor of 1.7 and 2.0, respectively.
Striving for further decrease in weight may lead to a thin-walled shaft. For instance, at diameters of do/di= 75/73 mm (wall thickness is of 1 mm), the weight of the hollow shaft is only 25% of that of the solid one. But the ratio of the wall thickness t to the mean radius r of the wall [r= (do +di)/4] is now r / t= 37. Because this ratio is bigger than 10, the shaft should be checked for buckling. In our case, the shaft is loaded with a torque only, so the critical (for stability in torsion) shear stress may be obtained from the following equation2:
As mentioned, r= 37 mm, t= 1 mm, so the critical stress is
The actual value of the shear stress
As we see, the actual shear stress is bigger than the critical one. That means that the hollow shaft with diameters 75/73 mm may buckle under load, and its dimensions should be changed so as to keep the actual shear stress not bigger than about half of its critical value.
Desirable changes in weight, stiffness, and the natural frequencies of a shaft may also be attained by changing the material properties, for example, by using titanium or aluminum alloys instead of steel. The permissibility of this replacement should be considered with respect to all elements of the shaft, such as splines, bearing necks, and others.